Performance analysis of
assisted-GNSS receivers
Nicolas Couronneau
Clare Hall
February 2013
A dissertation submitted for the degree of Doctor of Philosophy

Declaration
This dissertation is the result of work carried out in the Astrophysics Group of the
Cavendish Laboratory, Cambridge, between October 2009 and February 2013. The
work contained in the thesis is my own except where stated otherwise. No part of
this dissertation has been submitted for a degree, diploma or other qualification at
this or any other university. The total length of this dissertation does not exceed sixty
thousand words.
3

Acknowledgements
I would like to thank my supervisor, Dr Peter J. Duffett-Smith, for his enthusiasm and
friendship during my PhD. I have been extremely lucky to work with Peter. He gave
me a great deal of freedom during my research, while always boosting my confidence
to pursue my work when things did not work as expected. I have developed a lot
of my academic and professional skills during my conversations with Peter. He also
understood perfectly the “special needs” of a mature research student.
I want to thank Cambridge Silicon Radio for generously sponsoring my research.
The guidance and support I received from Stephen Goodwin, Nicolas Graube, Murray
Jarvis, and Ben Tarlow were very much appreciated and I would like to thank them
again for the time they spent with me. Ramsey Faragher and Sasha Mitelman were
instrumental in nurturing my interest in satellite navigation research and their expertise
was very much appreciated throughout my PhD work.
I am very gratefeul to Professor Gérard Lachapelle for inviting me to the Position,
Location, And Navigation group at the University of Calgary. My stay in the PLAN
group was relatively short but extremely valuable, in particular thanks to the expert
advice of Ali Broumandan, James Curran, and Tao Lin.
I had some great times at the Cavendish, in particular thanks to Alex and Michel
who have helped me stay motivated during these years.
I thank my college, Clare Hall, for contributing to my funding, which allowed me
to present at international conferences.
It is easy to forget how lucky one is to have a supportive and caring family, and I
would like to say «Merci Jean-Louis, Maryse, and Cécile» for all their efforts over the
years.
Finally, nothing would have been possible without the constant support, patience,
and love from my wife, Karen.
5

Summary
The goal of this thesis is to improve the understanding of the performance of Global
Navigation Satellite System (GNSS) receivers that use assistance data provided by
cellular networks. A typical example of such a receiver is a mobile phone including a
Global Positioning System (GPS) receiver. Using assistance data such as an accurate
estimate of the GPS system time is known to improve the availability and the time-to-
first-fix performance of a GNSS receiver. However, the performance depends on the
architecture of the cellular network and may vary significantly across networks. This
thesis presents three new contributions to the performance analysis of assisted-GNSS
receivers in cellular networks.
I first introduce a mathematical framework that can be used to calculate a theoretical
lower bound of the time-to-first-fix (TTFF) in an assisted-GNSS receiver. Existing
methods, for example the flow-graph method, generally focus on calculating the
theoretical mean acquisition time of a pseudo-noise signal for one satellite only. I extend
these methods to calculate the full probability distribution of the joint acquisition of
several satellites, as well as the sequential acquisition of satellites, which is commonly
performed in assisted receivers. The method is applied to real measurements made in
a multipath fading channel.
I next consider time assistance in unsynchronised cellular networks. It is often
argued that unsynchronised networks can not provide fine-time aiding since they do
not have a common clock, although few experimental results have been reported in the
existing literature. I carried out experiments on a GSM network, a second-generation
cellular network, in Cambridge, UK, in order to measure the time stability of the
synchronisation signals. The results showed a large variability in the time stabilities
across different base stations and I evaluated the performance of an ensemble filter that
combines the measurements into a single, more accurate, estimate of the universal time.
The main contribution is to show that the performance of such a filter is adequate to
provide fine-time assistance to a satellite navigation receiver.
7
Finally, I address the positioning performance of an assisted receiver in synchronised
cellular networks. Cellular positioning has been often investigated in the literature,
but few results on real networks have been presented. Many positioning methods
are proprietary and little information about their performance in real networks haven
been published publicly. A CDMA2000 cellular network in Calgary, Canada, was
used to collect experimental data. The time stability and the synchronisation of the
CDMA2000 pilot signals were excellent and were used to evaluate the performance of
CDMA2000-based cellular positioning system. I then developed a method to combine
the pseudo-range measurements from the GPS signals and the CDMA2000 base stations.
I evaluated the performance of positioning in both outdoor and indoor environments,
and I analysed the effects and the possible mitigation of non-line-of-sight signals. The
main contribution is to show that additional satellite navigation signals can improve
the accuracy of cellular positioning beyond what is theoretically expected from the
improvement in the geometry.
8
Contents
Declaration 3
Acknowledgements 5
Summary 7
Contents 9
1 Introduction to radio positioning systems 15
1.1 Fundamentals of positioning systems . . . . . . . . . . . . . . . . . . . . 15
1.2 Global navigation systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3 Positioning in cellular networks . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Estimation and detection in navigation receivers 35
2.1 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Signal detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Acquisition of GNSS signals . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Position estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Assisted-GNSS receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Analysis of the time-to-first-fix in assisted-GNSS receivers 59
3.1 Analysis of the acquisition of direct-sequence spread spectrum signals . 60
3.2 Mean features of assisted-receivers models . . . . . . . . . . . . . . . . . 68
3.3 Acquisition model of an assisted receiver . . . . . . . . . . . . . . . . . . 72
3.4 Illustration for typical signals . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5 Application to real signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.6 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . 89
9
4 Time assistance from unsynchronised cellular networks 93
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Overview of GSM signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 Oscillator models and time estimation . . . . . . . . . . . . . . . . . . . . 101
4.4 Time stability of the GSM air interface . . . . . . . . . . . . . . . . . . . . 105
4.5 Prediction of time deviations . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.6 Tracking of local oscillator bias . . . . . . . . . . . . . . . . . . . . . . . . 123
4.7 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5 GNSS aiding in synchronised cellular networks 135
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2 Measurements of times of arrival of CDMA2000 signals . . . . . . . . . . 138
5.3 Time stability and synchronisation of CDMA2000 networks . . . . . . . 144
5.4 Error analysis of the propagation time . . . . . . . . . . . . . . . . . . . . 148
5.5 Positioning using cellular signals only . . . . . . . . . . . . . . . . . . . . 157
5.6 Positioning using GNSS and cellular signals . . . . . . . . . . . . . . . . . 166
5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6 Conclusions 177
6.1 Summary of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
A GPS software receiver 181
A.1 Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
A.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
B CDMA2000 Sync Channel Message 185
B.1 Physical channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
B.2 Frames and super-frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
B.3 Sync Channel Message structure . . . . . . . . . . . . . . . . . . . . . . . 186
References 189
10
Nomenclature
C/N0 carrier-to-noise-density ratio
3GPP 3rd Generation Partnership Project
3GPP2 3rd Generation Partnership Project 2
AFLT Advanced Forward-Link Trilateration
AMPS Analogue Mobile Phone System
AWGN Additive White Gaussian Noise
BSC Base Station Controller
BSIC Base Station Identity Code
BTS Base Transceiver Station
C/A Coarse/Acquisition
CEP Circular Error Probable
CRLB Cramér-Rao Lower Bound
DFT Discrete Fourier Transform
DOP Dilution Of Precision
DSSS Direct-Sequence Spread Spectrum
ETSI European Telecommunications Standards Institute
FCC Federal Communications Commission
FDMA Frequency-Division Multiple Access
FFT Fast Fourier Transform
GLONASS Globalnaya Navigatsionnaya Sputnikovaya Sistema
GLRT Generalised Likelihood-Ratio Test
GMSK Gaussian Minimum Shift Keying
11
GPS Global Positioning System
GSM Global System for Mobile communications – initially Groupe Spécial Mobile
ICD Interface Control Document
IGS International GNSS Service
IP Internet Protocol
ITU International Telecommunication Union
LBS Location-based service
LMU Location Measurement Unit
LOS Line Of sight
LRT Likelihood-Ratio Test
LTE Long-Term Evolution
MS Mobile Station
NIS Normed Innovation Squared
NLOS Non Line Of Sight
OCXO Oven-Controlled Crystal Oscillator
PGF Probability Generating Function
ppb parts-per-billion
ppm parts-per-million
PRN Pseudo-Random Noise
PSTN Public Switched Telephone Network
RMS Root Mean Square
RSS Received Signal Strength
SAGE Space-Alternating Generalised Expectation-Maximisation
SCHM Sync Channel Message
SIC Successive Interference Cancellation
TCXO Temperature-Compensated Crystal Oscillator
TDMA Time-Division Multiple Access
TTFF Time-To-First-Fix
TTFH Time-To-First-Hit
12
UTC Coordinated Universal Time
UTM Universal Transverse Mercator
UWB Ultra-wide bandwidth, or ultra-wide band
13

CHAPTER 1
Introduction to radio positioning
systems
Radio positioning systems and cellular networks have changed our lives in ways that
may not have been possible to predict. Global navigation satellite systems (GNSSs),
and in particular the Global Positioning System (GPS), have brought positioning to
entry-level mobile devices, with an accuracy that was once only available to military
personnel and surveyors. Some of these devices, for example mobile phones, have also
access to the Internet using a cellular network. The combination of position information
and access to the Internet has paved the way to applications that are developed around
the user position. The new possibilities cover road charge payment, gaming, personal
navigation, etc.
1.1 Fundamentals of positioning systems
Several methods and systems have been developed in order to determine one’s position.
Some systems work in a local area while others operate globally; some rely on a
dedicated infrastructure while others work in an unprepared environment; some
work with radio-frequency signals while others rely on optical measurements, etc.
The classification of these methods is thus challenging, yet also desirable in order to
understand the landscape of possible positioning systems. We review some of the
most common technologies in this section, with a particular focus on radio-frequency
systems.
15
1. Introduction to radio positioning systems
Methods based on propagation time
Most radio-positioning systems measure the range between a transmitter and a receiver.
In the following, we assume that the measurements are performed between one receiver
and N transmitters; the analysis is also valid for the converse situation, with one
transmitter and several receivers. The user position, x, is related to its distance ri to the
transmitter i at position xi via the system of equations
r1 = ||x1 − x||
...
rN = ||xN − x|| ,
where ||v|| denotes the Euclidean norm of the vector v, i.e. ||v|| =
√
v2x + v2y for a vector v
or Cartesian coordinates [vx, vy]. We assume perfect measurements of time of flights in
this section and we address the case of noisy measurements in Section 2.4.
We now calculate the ranges from the measurements of propagation times. We
assume the signal is carried by an electromagnetic wave that propagates at the speed
of light in vacuum, c = 299 792 458 m/s. The signal is sent by transmitter i at time TOTi
and is received by the user at time TORi, from which we can calculate the range, ri,
using the relationship ri = c (TORi − TOTi). The receiver clock must be synchronised
with the transmitter in order to measure TOR in the same time scale as TOTi. The
signal carried by an electromagnetic wave “travels” about 3 m in 10 ns, hence the
time synchronisation between the receiver and the receiver must be of the order of
the nanosecond. Such a synchronisation accuracy is challenging and not feasible in
mass-market devices, specially when they operate in an environment with changing
temperatures. The synchronisation issue is a common problem in radio positioning
systems. It is, however, less problematic in systems with slower propagation speeds,
such as underwater Sonars or ultrasonic beacons that use acoustic waves.
We can modify the system of equations for the situation where all the transmitters
are synchronised together and the receiver operates in a different, local, time scale. The
modification consists of adding the user bias, bu, which is the time difference between
the time scale of the transmitters and the time scale of the receiver. We now have
ri/c = TORRXi − TOTi + bu, where TORRXi is the reception time in the receiver time scale.
The pseudo-range, ρi, is often used and is defined as
ρi = c
(
TORRXi − TOTi
)
.
The pseudo-range is more convenient to work with as it directly relates to the measure-
ments made by the receiver, in its local time scale. The previous system of equations
16
1.1. Fundamentals of positioning systems
can be written using the pseudo-range, giving
ρ1 = ||x1 − x|| − c bu
...
ρN = ||xN − x|| − c bu.
(1.1)
Equation 1.1 is often called the position equation and is fundamental in positioning
systems using radio signals. A common method to solve the position equation with
noisy measurements is presented in Section 2.4. The equation is presented in its
time-of-arrival (TOA) form, where all the pseudo-range times, ρi/c, are measured from
the origin of the receiver time scale.
Another possibility is to use only time-differences, which does not require the
receiver time scale to have a time origin – only time differences are required. This
approach is called time-difference of arrival (TDOA). The system of N equations to
solve is then 
ρ2 − ρ1 = ||x2 − x|| − ||x1 − x||
...
ρN − ρN−1 = ||xN − x|| − ||xN−1 − x|| .
(1.2)
The user bias is not an unknown variable as it does not appear in Equation 1.2, which
is normal since the TDOA relies on time differences only.
Both the TOA equation (Equation 1.1) and the TDOA equation (Equation 1.2) solve
the same problem and require the same number measurements. The TOA approach
is also known as trilateration and the TDOA approach as multilateration. The locii of
the positions in a two-dimensional TOA problem are circles and they are hyperbolas
in a TDOA problem. This name sometimes leads TDOA-based systems to be called
hyperbolic positioning. In Section 2.4, we will see that, with noisy measurements, the
solution to the TOA system of equations is generally more robust than for a TDOA
system.
However, the TDOA approach may be more practical in some applications. The
time differences can be easily calculated by the cross-correlation of two signals carrying
the same pattern without knowing the pattern beforehand. This situation arises when
locating the source of radio-frequency interference, for example. The content of the
interfering signal is generally unknown, however the same signal can be received from
different locations. The signals received at the various locations can be cross-correlated
in order to calculate the time differences, and the source of the interference can be
localised by solving the TDOA equation. The content of signals of opportunity, such as
17
1. Introduction to radio positioning systems
television and radio broadcasting, are generally unknown yet they are synchronised,
which makes them suitable for TDOA-based positioning. A receiver with a TOA
capability is more complicated since of local copy of the signal pattern is needed in
order to calculate the times of arrival.
The synchronisation of the transmitters is not always possible, in particular if they
have not been designed as a positioning system. Some cellular signals, for example,
carry voice and data to mobile phone users, without synchronisation between the
receiver and transmitter. A first possibility is to use dedicated equipment to measure
the time biases of the different transmitters, and send the corrections to the mobile, as
proposed by Duffett-Smith et al. (1992). However, dedicated equipments increase the
cost of the system. Methods such as Matrix, described by “Radio positioning systems”
(2003) and “Radio positioning systems” (2008), have been proposed to solve the problem
of unsynchronised receivers without relying on external equipment. The Matrix method
is described in Section 1.3. A system similar to Matrix has been proposed by Neven
et al. (2005) in order to synchronise several receivers for the automatic dependent
surveillance-broadcast (ADS-B) system, and air traffic monitoring system.
Other radio-frequency methods
We now present some radio-frequency techniques that do not rely on the measurement
of the signal propagation times. These alternative methods can be interesting in
mass-market applications since the accurate measurement of times can be technically
challenging.
Measuring the signal strength at the receiver location is one of the first radio-
positioning method designed. A path-loss model is used to calculate the signal power
as a function of the distance. A common path-loss model is P(d) = αP0d−n, where P(d)
is the power of the signal received at a distance d, for a transmitted signal of power P0;
α is a coefficient which is a function of the frequency and antenna gain; and n is the
path-loss exponent, which depends on the propagation environment. The accuracy of
the received signal strength (RSS) method is generally low. Uncertainties in the value
of the path-loss exponent and in the transmitted power, added to the time-varying
propagation conditions of the channel, affect the reliable estimation of the distance d. For
example, Alexander (1983) reports values of n between 1.4 and 6.5, a range of possible
values so wide that it leads to a large uncertainty in the range estimation. Nevertheless,
this method has enjoyed some popularity as the signal strength is generally reported
by devices, even if they have not been designed with a positioning capability. RSS is
thus potentially available on a large variety of platforms.
18
1.1. Fundamentals of positioning systems
Angle-of-arrival (AOA) methods use the angles between the user and some reference
points in order to calculate the position. These methods are sometimes called radio-
direction finding and the AOA position equation is solved by triangulation. Measuring
the angle generally requires a specially designed antenna and a phased-array antenna
is sometimes used. AOA methods are commonly utilised from positioning in cellular
networks. The radiation pattern of cellular networks antennas is often about 120°, which
leads to a division of each cell in 3 angular sectors. The base stations can then determine
the angles of arrivals of the signals from the mobile device with an uncertainty of
120°. Despite the large uncertainty in the arrival angle measurements, several base
stations may be in view of the same mobile device, thus increasing the accuracy of
the positioning. Navigation aids for aircraft also use AOA methods. Non-directional
beacons (NDB), for example, continuously transmit RF signals that are received by
an automatic direction finder (ADF) onboard aircraft. The pilot reads the bearings of
different beacons on the ADF, and then calculates his position by triangulation.
Fingerprinting is a method particularly suited for positioning with signals that
have not be designed for that usage. It is generally a two-stage process. An area is
first surveyed in order to record a “fingerprint” of signals, which is the summary of
the interesting features of a signal, such as the signal strengths, the waveform patterns,
etc. at known locations. The user then determines his position by finding a match
of the “fingerprints” of the received signals with existing records in the database.
Bahl et al. (2000) present one of the first evaluations of indoor positioning using the
fingerprinting of WiFi signals. They used the signal strengths of WiFi access points as
a fingerprint and they reported a median positioning error better than 5 m. Because
fingerprinting often relies of the received signal strength, it suffers from the same
problems as RSS techniques. Elnahrawy et al. (2004) present experimental results of
several fingerprinting algorithms. They highlight that neither increasing the sampling
resolution during the survey, nor improving the algorithms, leads to a significant
improvement in the position accuracy. These observations suggest that fingerprinting
methods can not achieve high accuracy, although such methods may be cost-effective
in some applications as they do not require modification of the infrastructure.
Finally, Cell-ID is a popular method of obtaining coarse positioning. It identifies
transmitters of known positions near a mobile station and calculates the position of the
mobile station, generally as a weighted average of the position of the transmitters. This
method is particularly common for positioning in cellular networks. It is similar to
fingerprinting but it is usually considered to be a separate technique, mostly because
it does not require surveying but it necessitates some knowledge of the network
19
1. Introduction to radio positioning systems
operations in order to identify the transmitters and their positions. On the other hand,
fingerprinting is generally possible using signals of opportunity, without knowing
the locations of the transmitters. Several databases have been built and contain the
identities and locations of cellular base stations. Cell-ID is a widespread method but it
is not as prevalent in the academic literature as some other methods, possibly because
it often relies on proprietary databases. The position accuracy is low; Trevisani et al.
(2004) measured average errors between 500 m in an urban centre in Italy and 1 km on
a highway in the United States.
Effects of radio propagation on positioning
Positioning with radio signals generally relies on the arrival times or the amplitude of
the signals. These two parameters are affected by the propagation conditions of the
radio waves. The degradation of radio signals is commonly the result of multipath
propagation, which is the propagation of radio waves along more than one path.
A first consequence of multipath propagation is multipath fading. The radio waves
that arrive at the receiver antenna from different paths are not necessarily in phase,
since each path introduces a different group delay. The recombination of radio waves
with random phases can be additive or subtractive, thus modifying the amplitude of
the resulting wave. Multipath fading has been studied extensively, in particular in
personal mobile communications, and several models of fading have been developed
for different environments. Hashemi (1993) provides a survey of theoretical models
and experimental results in indoor environments, where the most severe conditions of
radio propagation occur. The relative phase differences of the paths are variable in a
dynamic environment, hence the amplitude of the recombined wave at the receiver
varies with time. The path lengths are often affected by road traffic, surfaces moved
by the wind, etc. and, in practice, the fading is almost always time-varying. Fading
often occurs over distances which are of the order of the carrier wavelength. This
property can be exploited to mitigate the effect of signal fading by using spatial diversity
methods. These techniques consist of receiving the signal at more than location, using
more than one antenna, such that a subtractive combination at one location can be
compensated by an additive combination at another. Spatial diversity is often used in
radio-communications and it is also present in some GNSS receivers, in particular in
automotive applications. Multipath propagation can also change the polarisations of
the radio waves, hence antennas with different polarisation profiles are also used in
diversity methods.
Another consequence of multipath propagation is the degradation of the accuracy
20
1.1. Fundamentals of positioning systems
of the time measurements. We have seen previously that positioning systems generally
require the propagation times of the line-of-sight (LOS) signals in order to calculate the
position. The signal from the LOS path may not detected reliably in a multipath channel
if the combination of waves from non-line-of-sight (NLOS) paths produces a stronger
signal. This cause of degradation is often implied in the GNSS community when
multipath is mentioned. Multipath fading also affects GNSS receivers, in particular
during the acquisition of the signals, but it has less important consequences on the
positioning accuracy. Multipath propagation is notoriously difficult to mitigate. A
classical illustration is to consider the reception of satellite signals in urban centres
with tall buildings, a situation called an urban canyon. The multiple reflections of the
signals on the vertical surfaces of the buildings, combined with the masking of the
satellites, can lead to significant position errors for personal navigation. This problem
is addressed by Groves (2011), among others, who addresses the problem of locating
on which side of a street the user is.
Stand-alone positioning systems
Many radio-positioning systems are limited by their reliance on previously deployed
facilities, for example satellites or radio-frequency beacons. This dedicated equipment
has development, deployment, and maintenance costs which can outweigh the benefits
of bringing positioning capabilities. Radio-positioning systems also have physical
limitations. As an example, we will see in Chapter 5 how multipath propagation
and signal attenuation reduce the availability of radio-positioning inside buildings.
Similarly, positioning a robot underwater or in a mining site may not be possible
using radio-based methods. Additionally, a radio-positioning system may be subject
to interference, intentional or not, which may completely disrupt its availability. We
wish to develop positioning and navigation systems that can operate in unprepared
environments or that are not limited by the constraints of radio-propagation.
Vision has been frequently used in positioning technologies. Historically, sailors
used the positions of stars to determine their vessels’ positions, a technique then
adopted by early pilots. Celestial navigation is based on the observations of the angles
of known stars, using a sextant for example. The position is then calculated using
a star almanac and the time of the measurements. Celestial navigation is seldom
used nowadays, except to determine the attitude – not necessarily the position – of a
spacecraft. Visual odometry, another vision-based method, consists in estimating the
motion of a camera by tracking the motion of visual features over time. The trajectory of
the features gives the mobile’s velocity vector, which is then integrated to calculate the
21
1. Introduction to radio positioning systems
position. Some amount of diversity in the images is required in order to find enough
features with a geometry that allows the calculation of the motion. It is also possible to
use known features or recognize objects in order to obtain absolute positions. Vision
is often used in robotics for local path planning and obstacle avoidance, as used by
Thrun et al. (2006) during the navigation “Grand Challenge” of the American Defense
Advanced Research Projects Agency (DARPA). It is not a positioning task as such,
although it can provide location information of obstacles in a local area.
An active area of research concerns the navigation of a mobile in an unprepared and
unknown environment. Known as simultaneous localisation and mapping (SLAM),
these techniques contrive to build a map of unknown surroundings at the same time as
positioning the mobile in it. It is, in general, a highly computational problem and many
techniques have been proposed to simplify this task. Newman et al. (2009) present the
results of a visual SLAM system that achieves a position error better than one metre.
Recently, Faragher et al. (2012) proposed a method of SLAM in an office building using
radio-frequency signals of opportunity, accelerometers and magnetometers from a
smartphone, and a foot-mounted accelerometer.
Inertial navigation systems were one of the first standalone solutions developed.
They are based on the measurement of the acceleration vector and the rotation rates
of the user. These derivatives of the user motion can be integrated to calculate the
user’s change of position. Inertial systems were initially expensive and reserved to
specific applications, such as aircraft and missile guidance. Microelectromechanical
systems (MEMS) have lowered the cost of inertial sensors and inertial measurement
units (IMUs) are now gaining popularity in commercial products. Inertial systems are
commonly found in commercial and military aircraft, ships, and submarines.
Sensor fusion
The measurements from different systems are often combined in order to improve the
position accuracy. Inertial measurements units, for example, are affected by the biases
of their sensors, which create a drift of the user position as the biases are integrated. For
this reason, IMUs are commonly supplemented with GNSS measurements. The GNSS
signals bring high accuracy and global positioning, whereas the IMU measurements
bring robustness to the system when the RF signals become unavailable. Inertial
positions can also be calculated at a higher rate than GNSS positions, hence the coupling
of an IMU and a GNSS receiver is particularly attractive for platforms with large
dynamics. Similarly, the combination of Cell-ID, fingerprinting, and RSS can be used to
22
1.2. Global navigation systems
estimate the position of a cellular device without changing the network infrastructure
or the receiver. This approach is often used for positioning on smartphones.
Sensor fusion is a major topic in navigation systems research. It is based on Bayesian
probabilistic methods to combine noisy information from several sensors. A major
building block of sensor fusion is the method developed by Kalman (1960), now
commonly called Kalman filter, which is presented in Chapter 2. There are several other
tools inspired by the original Kalman filter, which can deal with non-linearities and
non-Gaussian noises, for example. Some challenges of modern sensor fusion include
the characterisation of the sensor noises and failure modes, the modelling of the mobile
platform dynamics, the development of efficient algorithms for real-time operations on
mass-market devices, etc. I have used an approach similar to sensor fusion in Chapter 4,
in order to improve the time stability of a receiver.
1.2 Global navigation systems
History
Radio positioning systems were developed in the 1940s to help the positioning of ships
and aircraft. One of the first global systems was the LOng RAnge Navigation, LORAN-
A, system. Synchronised transmitters periodically sent radio-frequency signals in a
low-frequency band, at about 100 kHz. The user measured the time differences between
the arrival times of pairs of signals and used these to calculate his position. The typical
accuracy was about 200 m, although this could be degraded inland. The LORAN-A
system was extended to form the LORAN-C systems (United States Coast Guard, 1994),
which was used for marine and air navigation until it was decommissioned in 2010 in
the United States. The General Lighthouse Authorities (GLA) of the United Kingdom
have promoted eLoran (enhanced-LORAN), which improves the performance of
LORAN-C and facilitates its integration with satellite navigation systems. The OMEGA
system (Kasper et al., 1978) was a radio navigation system whose principles were
similar to LORAN-C. It used very low-frequency signals, between 10.2 and 13.6 kHz,
which provided worldwide coverage but had less accuracy than LORAN-C. It was
decommissioned in 1997 by the United States. DECCA was a British navigation system
similar to LORAN that was developed during World War II. Stations transmitted a
continuous signal at low frequency and the user measured the phase differences of
the RF signals. The system was shut down in 2000. LORAN, OMEGA, and DECCA
operated using the measurements of time or phase differences and therefore were
hyperbolic positioning systems.
23
1. Introduction to radio positioning systems
Terrestrial systems such as LORAN, DECCA, and OMEGA, have shortcomings.
The transmitters are located in the same horizontal plane, a geometry that can not
provide useful altitude information. The horizontal propagation of the radio waves also
encounters several obstacles inland and is affected by built-up areas, which degrades the
position accuracy. Furthermore, low-frequency signals are better suited for long-range
propagation, but high-frequency signals provide better accuracy.
These shortcomings led to the development of satellite navigation systems. The
first satellite navigation system was the United States Navy Navigation Satellite System
(NNSS), more commonly known as Transit. A user would monitor the Doppler shift of
the Transit radio-frequency carrier in order to calculate his position. The constellation
was small and only one satellite was in view of the user in general. This geometry
did not allow the calculation of the position immediately and the user often had to
wait until enough Doppler measurements had been collected from different points of a
satellite orbit before the position equation could be solved. The American Department
of Defense (DOD) formed a working group in 1973 and started developing what became
the Global Positioning System (GPS).
The Global Positioning System
The Global Positioning System is arguably the most commonly-used global navigation
satellite system (GNSS). It operates by measuring the propagation times of radio-
frequency signals sent by satellites. The GPS satellites continuously broadcast signals
whose transmission times are synchronised across all satellites. The user can then
calculate his position using the TOA method. The time stability and synchronisation
of the transmitted signals is excellent, thanks to the use of atomic clocks onboard the
satellites and monitoring by ground-based stations that upload time corrections to the
satellites. The high stability of the signals is exploited by many timing applications that
do not always require the positioning capabilities of GPS, for instance to synchronise
equipments separated by large distances.
The Global Positioning System is described in several references, including the
books by Parkinson et al. (1996) and the website of the National Coordination Office
for Space-Based Positioning, Navigation, and Timing (2012). The ground segment is
operated by the United States Air Force and is composed of several command and
monitoring stations across the world. The space segment comprises a constellation of
satellites that ensures worldwide coverage – between 24 and 31 operational satellites in
general. The satellites orbits are quasi-circular, with a radius of 26 000 km, resulting in
an orbital period of 12 h in sidereal time. The signals are transmitted using a direct-
24
1.2. Global navigation systems
sequence spread spectrum (DSSS) technique. This method modulates the original
signal with a pseudo-random noise (PRN) sequence. The binary elements of a PRN
sequence are called chips. They are often generated by a linear feedback shift register.
GPS satellites transmit two signals with different PRN characteristics. The coarse
acquisition (C/A) code is transmitted on the frequency link 1 (L1), whose carrier
frequency is 1575.42 MHz. It has a chipping rate of 1.023 Mchips/s and a length of
1023 chips, thus the period of the C/A code, TCA, is 1 ms. The precision (P) code is
spread at a ten times faster chipping rate. It is transmitted on the L1 frequency, in
quadrature to the C/A code, and also on the frequency link 2 (L2), at 1227.60 MHz. The
P code is encrypted to form the Y code, often called the P(Y) code. The encryption
key is provided by the DOD to authorised users only. The C/A code was intentionally
degraded by the selective availability (S/A) feature until May 2000.
The use of DSSS has several advantages. The modulation distributes the power
of the original signal across a large spectrum, and the maximum power spectral
density can even be below the noise floor of the receiver. This has the advantage of
covert operation and it limits the electro-magnetic interference of other equipment, in
particular radio-telescopes. The DoD guarantees the L1 C/A signal is received with
a strength of at least −183 dB W at the surface of the Earth. This brings the power
spectrum density below the noise floor, as illustrated in Figure 1.1.
−10 −5 0 5 10
Frequency centred on L1 (MHz)
−250
−240
−230
−220
−210
−200
PS
D
(d
BW
/H
z)
Noise floor
Figure 1.1: Illustration of the power spectrum density of the L1 C/A signal. The
noise floor is set to −204 dB W/Hz, a value typical of a terrestrial GPS receiver.
The DSSS modulation also facilitates the measurement of the arrival times. A PRN
sequence can be built such that its auto-correlation function has all the signal energy
concentrated at a lag of zero, and negligible correlation at any other lag. The receiver
can recover the arrival time by correlating the received signal with a local replica of the
PRN sequence. The correlation properties of a PRN sequence is illustrated in Figure 1.2.
We note that the correlation function becomes zero almost all the time for any distance
25
1. Introduction to radio positioning systems
large than 300 m from the origin.
−3 −2 −1 0 1 2 3
Distance-lag from origin (km)
0
500
1023
C
or
re
la
ti
on
va
lu
e
(a
m
pl
it
ud
e) −10 −5 0 5 10
Lag in chips from origin (chips)
Figure 1.2: Autocorrelation function of the PRN 1 sequence of the C/A code. The
correlation values are in the amplitude domain.
Finally, the DSSS modulation provides code-division multiple access (CDMA), a
technique that allows several transmitters to share a common radio-frequency carrier
without interference. This is achieved by choosing a class of PRN sequences that
have little cross-correlation between each other: all the sequences are orthogonal to
each other. The family of Gold codes (Gold, 1967) has this property and is used by
the GPS L1 C/A signal. All the satellites can transmit on the same frequency and
each satellite is identified by its unique PRN code generator. Figure 1.3 illustrates the
comparison of the first 100 chips of the PRN 1 and 2 of the C/A code. It can be shown,
as in Parkinson et al. (1996, chapter 3), that the worst-case of cross-correlation power is
about −23.8 dB below the nominal correlation power.
0 20 40 60 80 100
-1
+1
PR
N
1
0 20 40 60 80 100
Time (µs)
-1
+1
PR
N
2
Figure 1.3: First 100 chips of the PRN 1 and PRN 2 sequences of the C/A code.
The signals also carry the GPS navigation message, which is modulated by a
binary-phase shift keying modulation at 50 bits/s. The navigation message contains
information needed to determine the satellite positions and the times of transmission of
the signals with high accuracy. This includes the ephemeris data and the time-of-week
26
1.2. Global navigation systems
(TOW) counter, which is gives the time of transmission of the signal in the GPS system
time scale. The ephemeris data provide the orbital elements and correction terms
that are required to calculate the position of the satellite at a given time, as well as
information about the time deviations of the atomic clocks. The parameters given by
the ephemeris are only valid for a short time interval, generally about four hours. The
almanac data provide the orbital elements and correction terms with less accuracy
than the ephemeris data, but they are valid for several days. The descriptions of
the navigation message, the C/A code generators, and the signal specifications are
described in GPS interface control document (ICD, GPS Joint Program Office, 2004).
Other Global Navigation Satellite Systems
The GPS L1 C/A signal is undoubtedly the most frequently used GNSS signal at this
time. Other nations have developed their own satellite-based systems and these are
gaining increasing popularity among receiver manufacturers.
The Russian GLObal NAvigation Satellite System, GLONASS, has been developed
initially by the Soviet Union and then by Russia (Russian Institute of Space Device
Engineering, 2008). The satellites are identified by a their frequencies, instead of a
PRN sequence as in the GPS, which creates integration problems in GPS receiver,
since the frontends need a wider bandwidth to accommodate both the GPS signals
on a fixed carrier frequency and the GLONASS signals at variable central frequencies.
Nevertheless, the system is fully operational and its 31 satellites, as of November 2012,
offer global coverage. GLONASS broadcast pseudo-noise signals in frequency bands
close to those of GPS.
The European Union has been working on the Galileo system (European Union,
2012) and has deployed four in-orbit validation satellites as of November 2012. Galileo
has been designed with interoperability with GPS in mind. The Compass system,
also called Beidou-2, is being developed by the People’s Republic of China. There are
currently 16 Compass satellites, but most of the interface control document has not
been made public. The GPS system is also going through an upgrade with the addition
of new navigation signals. A new civilian signal on the L2 frequency, the L2C signal,
has a faster spreading rate than L1 C/A, which can improve the position accuracy.
The presence of two civilian signals at different carrier frequencies also facilitates the
correction of ionospheric errors.
Satellite augmentation systems provide information that improves the performance
of a GNSS receiver, for example the status of a GNSS constellation, the ephemeris, and the
clock and ionospheric correction data. The Wide-Area Augmentation System (WAAS)
27
1. Introduction to radio positioning systems
and the European Geostationary Navigation Overlay Service (EGNOS) are currently in
use to broadcast information about the GPS constellation and the GPS L1 C/A signal.
The signals are transmitted by geostationary satellites on the L1 frequency.
The current trend is towards receivers capable of operating with signals from
multiple systems on several frequencies. Receivers compatible with GPS, Galileo,
and Glonass, and operating with signals from two different frequency bands, are
commercially available as of 2012.
1.3 Positioning in cellular networks
The wide majority of cellular networks provide the infrastructure for voice and data
communications using mobile phones. They have changed the nature of communication
between individuals in depth and over a short period of time. The United Kingdom
had some 81.6 million mobile subscribers in 2011 according to Ofcom (2012), for a
population of 62 million people. Cellular networks originally carried voice only, but
mobile devices now offer data services, such as access to the Internet. At the same time,
the computing power of the devices has increased and applications that require intense
data processing are now widely available on personal handsets. This situation opened
new possibilities for positioning capabilities on mobile devices. In this section, we
review the main technologies of cellular networks and their positioning possibilities.
History and technology of cellular networks
Different technologies exist and all have in common the division of an area into many
cells, each cell being covered by a base transceiver station (BTS). The Shannon-Hartley
theorem shows that a communication channel with a given bandwidth and signal-to-
noise ratio, such as the radio link between the BTS and the user, has a finite capacity.
This can be a problem as the number of users increases in a cell, thus requiring additional
capacity. The sizes of the cells change depending on their locations so that the channel
capacity is optimised. Other constraints, such as the propagation conditions and
technical limitations of the network, also affect the sizes of the cells.
The first cellular systems relied on analogue signals to communicate. The analogue
mobile phone system (AMPS), for example, was introduced in the early 1980s, mostly in
North America and Australia. The total access communication system (TACS), a variant
of AMPS, was used in Western Europe. The deployment and adoption of these early
systems was limited and I could not find evidence of any of these networks being still
operational in 2012. The second-generation (2G) networks were arguably more popular.
28
1.3. Positioning in cellular networks
They relied on digital radio communications, with two major technologies dominating
the market. The global system for mobile communications (GSM, although originally
standing for Groupe Spécial Mobile) is the most common technology in 2G networks.
The specifications are maintained by the European telecommunications standards
institute, (ETSI, ETSI website), and are generally published by the 3rd Generation
Partnership Project (3GPP, 3GPP website). A major feature of this network is that the
BTSs need not be synchronised. Each BTS is attributed a set of unique frequencies
in its area, hence implementing frequency-division multiple access (FDMA) over the
entire network. The users, also called mobile stations, communicate with the BTSs over
a channel regulated by time-division multiple access (TDMA). The communications
occur on a common frequency and at specific time intervals, such that the signals from
two mobile stations do not overlap. GSM networks cover more than 90 % of the world’s
population according to the GSM Association, (GSM Association website), a group of
companies supporting the GSM standard.
The alternative to GSM among second-generation networks is the IS-95 standard,
marketed as cdmaOne. The standard is maintained by the telecommunication industry
association (TIA) and is not freely available. One of the main characteristics of the
standard, and its biggest architectural difference with GSM, is the requirement for
synchronised BTSs. The standard does not have a FDMA component like GSM and
all base stations within a network transmit on the same frequency. The absence of
interference and the access to the BTSs are assured by code-division multiple access.
The signal from each BTS is modulated by a pseudo-noise sequence, similarly to the
DS-SS modulation of GPS signals. The base stations are identified by the time offsets
of their spreading sequences. This implementation requires a common time scale for
all the transmitters, to mark the time origin of the time offsets. The IS-95 standard
specified the GPS time scale is to be used, which in practice requires a GPS receiver in
time-transfer mode at every BTS site.
The third-generation (3G) standards of cellular communications provide larger
bandwidths and better support of data services. They are also divided into two
main competitors: the Universal Mobile Telecommunication System (UMTS) has been
developed by the 3GPP to succeed to the GSM standard, while the CDMA2000 standard
has been built on the foundation of IS-95. UMTS has several differences with the GSM
standard. The air interface, which specifies the radio communications between the
BTSs and the mobile stations, has two different implementations. The most common
is the W-CDMA standard, which is based of code-division multiple access and has a
bandwidth of about 5 MHz. The network architecture is, like GSM, unsynchronised,
29
1. Introduction to radio positioning systems
despite the use of CDMA. The CDMA2000 standard is maintained by and publicly
available from the 3rd Generation Partnership Project 2 (3GPP2, not to be confused with
3GPP).
The fourth-generation systems are being deployed but are not as widespread as
3G networks as of 2012. The main technology is the Long-Term Evolution (LTE)
standard (3GPP - LTE), which affords a larger bandwidth than UMTS. LTE employs
orthogonal frequency-division multiple (OFDM) access.
Cellular positioning techniques
The communication industry quickly became interested in knowing the position of
mobile device users in cellular networks. User positioning became the cornerstone of a
new range of applications, known as location-based services (LBSs). The position is
generally presented to the user of the mobile terminal, for example by presenting a map.
Other LBSs transmit the position to a third-party, for example to locate an emergency
call, to restrict some services to a given geographical area (geo-fencing), or to collect
road toll charges. The possibility of obtaining the user’s position without consent, or
even knowledge, has raised privacy concerns, as illustrated by Malte Spitz.
A major driver of research in the positioning of mobile phones is the enhanced-911,
or E911 mandate of the Federal Communications Commission (FCC) of the United
States. This mandate requires that mobile network operators provide the user position to
emergency services during an emergency call (9-1-1 in the United States). The position
accuracy requirements are determined by the FCC. They vary with the environment,
but are generally in the range of 50 to 150 m. These accuracy targets created a sustained
interest in developing accurate and robust positioning methods for cellular devices.
Various solutions have been proposed and I review some of literature around this topic.
A significant part of the research into positioning methods for commercial devices is
performed by private companies and is not always published in the academic literature.
How the position of the user is calculated depends on the peripherals available on
the device. We review some of the most common methods and we refer to Gustafsson
et al. (2005) for a more complete treatment. Klukas (1997) investigated a TOA positioning
system for an AMPS network. The network was not synchronised and the author used
GPS receivers in order to calibrate the transmitters’ clocks. The multipath propagation
was mitigated by using the multiple signal classification (MUSIC, Schmidt, 1986)
algorithm. The reported root mean square error (RMS) was between 190 m and 290 m
in an urban environment.
30
1.3. Positioning in cellular networks
Positioning with GSM networks is more challenging because the network is not
synchronised. Most methods rely on dedicated equipment within the cellular network
to measure the time deviations of the signals from the base stations. A solution is
the enhanced-observed time difference (E-OTD) of arrival, also called observed time
difference of arrival (OTDOA) in UMTS networks. It relies on the presence of location
measurement units (LMUs) in the network, which continuously measure the biases of
the signals from the BTSs. Drane et al. (1998) attained a RMS accuracy of about 150 m
in an urban environment, where the timing signals were calibrated.
LMUs need to be located at several points of the network in order to maximise the
coverage of the base stations. The deployment and maintenance costs of LMUs can
be problematic for network operators. The Matrix system, previously mentioned, is a
solution that does not rely on dedicated network equipment. The Matrix technique
works in cellular networks, where M receivers at different locations can measure the
arrival times of the signals transmitted by N base stations. This method exploits the
large number of receivers in common view of a set of transmitters in order to find
the unknown time biases of the transmitters. It is then possible to write a system
of equations with the biases of all the clocks in the network, both at the receiving
terminals and the serving BTSs. The new system of equations can be solved under
the condition M N ≥ 3M + N − 1. Duffett-Smith et al. (2001) present the results of
experiments in a dense urban environment. The 67-percentile position errors are
about 130 m, which is comparable to the GPS errors measured at the same locations
in that challenging environment. The advantages of Matrix over GPS are the better
availability of the cellular signals and their faster acquisition times. An extension to the
Matrix method, called the Solo Matrix system, regularly performs a time calibration
of the GSM transmitters using the position and time of a GPS receiver. The position
is then calculated using a TDOA approach, assuming the transmitter clocks have not
changed.Duffett-Smith et al. (2004) show that the short-term time stability of GSM
networks is adequate for outdoor positioning without GPS measurements. Further
work in the area of cellular network stability is presented in Chapter 4.
Positioning in synchronised networks, such as CDMA2000, is generally simpler
than for unsynchronised network, since the transmitters are synchronised and a TDOA
or TOA approach can be used. Caffery et al. (1998b) provide a comprehensive overview
of the methods available for positioning in CDMA cellular networks. Positioning in
CDMA2000 is the central topic of Chapter 5.
Early systems used Cell-ID to determine the mobile of the user terminal. The
identity of the nearest base station is used as the position of the mobile. AOA is
31
1. Introduction to radio positioning systems
generally available as well, since most cells are divided into sectors, with a different
transmitter identity in each sector. Cell-ID and AOA methods demand some knowledge
about the identity, coverage pattern, and positions of the BTSs. This information is
generally not available, unless a company which develops the positioning technology
has entered an agreement with the network’s operator. Alternatively, the developer
of a positioning method can perform a survey of the operator’s network. The most
cost-efficient method to do so is to exploit the user’s handsets as measuring equipment,
an approach which is possible only if the technology developer already has a access to
a large number of users.
Assisted-GNSS receivers
Most of the radio-positioning systems based on the signals from cellular base stations
have difficulties reaching the requirements set by the FCC. These difficulties, combined
with the increasing affordability of GNSS receiver chips and the demand for more
accuracy from LBSs, have favoured the installation of GNSS receivers in cellular devices.
The performance of these receivers can be improved by using external information
accessible via the cellular data connection. The cellular network thus brings assistance
information to the GNSS receiver. I explain the operation of an assisted-GNSS receiver
in Section 2.5.
1.4 Thesis outline
Assisted-GNSS receivers have different architectures and operations from conventional
receivers. The performance of conventional receivers has been investigated extensively
in the existing literature, in both theoretical and experimental studies. However, the
performance of assisted receivers has not received the same attention. Furthermore,
the performance of an assisted receiver depends on the performance of the cellular
network, which is generally not addressed in the literature.
The central topic of this thesis is the analysis of the performance of assisted-GPS
receivers. I analysed the acquisition performance from a theoretical point of view and I
evaluated the characteristics of the cellular networks through experiments. The plan of
this thesis follows the chronology of my work. I first developed an analytical framework
to evaluate the theoretical time-to-first-fix of assisted-GPS receivers, which is presented
in Chapter 3. I regulated the theoretical model with experimental measurements in
order to analyse the minimum bound of the time-to-first-fix in a real environment. I
then focused on the quality of the assistance data provided by a real GSM cellular
32
1.4. Thesis outline
network. I improved the stability of the timing signals and the results of this work
are detailed in Chapter 4. Finally, I evaluated a method to position a handset in a
CDMA2000 network. The results of these experiments are described in Chapter 5.
33

CHAPTER 2
Estimation and detection in
navigation receivers
The acquisition of a radio-frequency signal is one of the main topics of this thesis.
We will encounter signals coming from GPS satellites and GSM and CDMA2000 base
stations. These signals need to be received and decoded, operations which are better
understood in terms of parameter estimation and signal detection. These concepts have
been studied in several textbooks and I focus on the aspects relevant to this thesis. The
theoretical foundations are treated in great detail by Van Trees (1968) and Kay (1993),
while Pany (2010) provides more information about the application of these principles
to GNSS receivers.
I introduce some theoretical principles of signal estimation and detection, then I
apply these results to the case of pseudo-noise signals. Pseudo-noise signals are used
in most GNSS constellations and in the CDMA2000 cellular network that is the subject
of Chapter 5. We also review the estimation problem in dynamic linear systems and
we introduce the Kalman filter. The work presented in this thesis is focused on the
performance of assisted receivers and I discuss the architecture and operations of this
type of receivers at the end of this chapter.
Estimation theory generally belongs to either one of two different branches of
probability theory: the Bayesian approach or the frequentist approach. The Bayesian
approach is generally preferred as it avoids many of the pitfalls found in frequentist
methods, as discussed in detail in Jaynes (2003). However, the frequentist approach
is commonly encountered in signal processing, as it is easier to implement and does
not bring a significant degradation of the results when compared with the Bayesian
35
2. Estimation and detection in navigation receivers
approach. I used maximum-likelihood methods, which are frequentist, to estimate the
parameters of digital signals such as the baseband signal of a radio receiver. Bayesian
methods were reserved for the processing of higher-level data, such as the receiver
position estimates or the time deviations of an oscillator.
2.1 Parameter estimation
We seek to estimate the unknown values of the parameters of a probabilistic model
representative of a real system. Observation of the physical system gives measurements,
from which we can infer the values of the model parameters. Parameter estimation is not
concerned about whether the observations are compatible with the model or whether
another model could explain the observations better. This task will be performed in the
detection stage, which we study in Section 2.2.
We first introduce some general concepts about estimation and probability theory.
We then address the general problem of estimating the parameters of a signal in noise
and we study in more detail the case of pseudo-noise signals. Finally, we look at the
estimation of parameters in the case of a dynamic linear model and introduce the linear
Kalman filter. This method will be used in Chapters 4 and 5, when filtering the timing
measurements from cellular base stations.
Maximum-likelihood estimation
The estimation process produces information about the plausible values of the vector
of parameters, θ, of the model under consideration. The system is observed and the
measurement are denoted D. All the knowledge about the probabilistic model is
enclosed in the likelihood function, p(D|θ), which describes the relationships between
the model, the parameter vector, θ, and the observations,D. The parameter vector is
the variable of the likelihood function and the observation vector,D, is a constant. It is
often convenient to rewrite the likelihood function as
L(θ|D) = p(D|θ),
which highlights the dependence of the parameter vector and the fact that the likelihood
function is not a probability density function.
How the inference about the parameters is performed depends on the approach to
probabilities one takes. The frequentist approach treats the parameters as unknown,
yet non-random, variables. Estimation using frequentist methods gives a single point
36
2.1. Parameter estimation
estimate of the parameter, called the maximum-likelihood (ML) estimate, which is defined
as
θˆ = arg max
θ
L(θ|D).
Bayesian methods, on the other hand, treat the parameters as random variables
with known prior probability distributions. The result of the inference is contained in
the posterior distribution of the parameters. The Bayes theorem gives the relationship
between the prior distribution, of probability density function p(θ), and the posterior
distribution, p(θ|D), as
p(θ|D) = 1
K
L(θ|D) p(θ),
where K is a normalising constant, also called evidence or marginal likelihood. More
formally, K =
∫
θ
L(θ|D) p(θ) dθ.
Bayesian inference thus does not produce a single point estimate of the parameter
values. The information is contained in the entire posterior distribution. However,
we often need a point estimate in order to make decisions. Berger (1985) discusses
how the loss functions, which can be subjective and depends on the decision problem,
affect the function used to generate the point estimate. This discussion is not necessary
for this thesis, yet it provides some useful insight about the meaning of statistical
estimators. The main points are that the maximum-a-posteriori (MAP) estimate,
θˆMAP = arg maxθ p(θ|D), corresponds to the maximum-likelihood estimate with a
uniform prior distribution; the posterior mean corresponds to a quadratic loss function;
and the posterior median corresponds to a linear loss function.
General signal in noise
We first review the maximum-likelihood estimation of parameters with a particular
emphasis on radio signals. For now, we consider a general complex signal x(t,θ) where
θ is the parameter vector of interest and t is the time. This signal is transmitted over an
additive white Gaussian noise (AWGN) channel.
This representation is typical of the equivalent baseband signal in a radio-frequency
receiver. Proakis (2000), among others, reviews the differences between passband and
baseband representations. The Gaussian noise introduced by the channel is represented
at baseband by a Gaussian random process ν(t) whose covariance function has the
following properties: 〈ν(t) ν∗(t′)〉 = N0 δ(t − t′), where δ is the Dirac function and
N0 is the dual-sided noise power spectral density. The received signal, z(t), is thus
z(t) = x(t,θ) + ν(t).
37
2. Estimation and detection in navigation receivers
The continuous-time baseband signal is then sampled and converted to a discrete
signal. The sampling period is Ts and the sampling frequency, fs, is fs = 1/Ts. The
sampling instants, tk, are thus tk = kTs. M samples of the received signals are represented
by the vector z1:M = [z1, . . . , zM], with
zk = x(tk,θ) + νk,
and νk is a sequence of independent, complex, Gaussian random variables. The
sequence νk represents the noise after sampling. The mathematical representation
of a stochastic process, such as the noise process ν(t), after the sampling operation
requires some attention. The statistical properties of the complex noise sequence νk are
now 〈νk νk′〉 = σ2 δk−k′ , with δk is the Kronecker delta function. The noise variance is
σ2 = N0 fs and we neglect the quantisation noise.
The likelihood function of the parameter vector θ in the presence of a signal and
given the observed signal z1:M is
L(θ|z1:M) =p(z1:M|θ) = 1(piσ)M e
− 1
σ2
∑M
k=1|zk−x(tk,θ)|2 . (2.1)
The logarithm of the likelihood function, or log-likelihood, is more commonly used.
logL(θ|z1:M) = − 1
σ2
M∑
k=1
|zk − x(tk,θ)|2 + K
= − 1
σ2
M∑
k=1
[
|zk|2 + |x(tk,θ)|2 − 2 Re
(
x(tk,θ)z∗k
)]
+ K,
where K = −N/2 log(piσ2).
This relationship can be understood by introducing the energies, divided by the
sampling period, contained in a signal of length M samples. The energy of the signal of
interest is Ex(θ) =
∑M
k=1 |x(tk,θ)|2 and the energy contained in the received signal, z1:M,
is Ez =
∑M
k=1 |zk|2. The log-likelihood can then be written as
logL(θ|z1:M) = K − Ex(θ) + Ez
σ2
+
2
σ2
Re
 M∑
k=1
x(tk, θ)z∗1:M
. (2.2)
The log-likelihood can be interpreted geometrically: if we consider the vectors
containing the theoretical signal response x(θ) = [x1(θ), . . . , xM(θ)] and the vector of
observations z1:M, then we can express the log-likelihood in term of the dot product
of the two vectors: logL(θ|z1:M) = 2 Re (x(θ) · z∗). The algebraic representation leads
to a geometric interpretation: the maximum of the likelihood is reached when the
parameters are adjusted so that the vector of the received measurements is parallel to
the vector formed by the samples of the theoretical signal sent.
38
2.1. Parameter estimation
Estimation of pseudo-noise signals
The signal model is now set to represent a pseudo-noise signal, typical of a direct-
sequence spread-spectrum signal used by GNSSs. The continuous-time signal of
interest has the form
x(t, a, τ, f ) = a 1(t − τ) e j2pi f t, (2.3)
where a is the complex amplitude of the baseband signal, 1(t) is the spreading code
sequence, τ is the code-phase delay and f is the residual frequency in the baseband
signal. The residual phase offset is included in the complex amplitude a. The power
of the signal modelled by Equation 2.3 is constant and, therefore, Ex(θ) = N|a|2. One
should remember that this is the power of the baseband signal – the power of the
signal-in-space of the GNSS satellites, C, which is reported in the ICD, is generally
given for the RF carrier signal, which is half the power of the baseband signal, C = |a|
2
2 .
The log-likelihood function of the baseband signal can be derived from Equation 2.2
and written as
logL(θ|z1:M) = K − N|a|
2 + Ez
σ2
+
2
σ2
Re
 M∑
k=1
a 1(tk − τ) e j2pi f tk z∗k
 . (2.4)
The parameters of interest are now θ = (a, τ, f ). In a GNSS receiver, τ is related to
the pseudo-range and f to the frequency offset, which is caused by the Doppler effect
and the frequency bias of the local oscillator. These two parameters are often called
the synchronisation parameters in the spread-spectrum communication literature. The
amplitude, a, is generally treated as a nuisance parameter, which we do not seek to
estimate explicitly. Pany (2010), among others, shows that the maximum-likelihood
estimate of the amplitude, aˆ, can be calculated as
aˆ(τ, f ) =
1
N
M∑
i=1
zi 1∗(ti − τ) e− j2pi f ti . (2.5)
We can then replace the amplitude a by its estimate aˆ in the likelihood function, a
substitution that does not affect the results because of the invariance principle applied
to maximum-likelihood estimation. The expression also highlights the importance of
the correlators to detect and estimate the signal parameters. A correlator of length M
performs the operation
Sn(τ, f ) =
n+M+1∑
k=n
zk1∗(tk − τ) e− j2pi f tk , (2.6)
where n is the index of the first sample of the signal to be correlated. The correlator
is, intuitively, an operation that matches the received signal with the conjugate of the
39
2. Estimation and detection in navigation receivers
theoretical signal. This operation is also called matched filtering. The matched filter is
commonly found in the signal estimation literature, in particular for the processing of
Radar signals. The matched filter can also be derived directly by calculating the filter
that maximises the signal-to-noise ratio at the receiver. The derivation I presented here
highlights the fact that the amplitude parameter is treated as a nuisance parameter,
and that better knowledge about the distribution of the signal strength may lead to a
different estimator.
The log-likelihood function of the general estimation problem, given in Equation 2.2,
can now be written as
logL(τ, f |z1:M) = K −
M
∣∣∣aˆ(τ, f )∣∣∣2 + Ez
σ2
+
2
σ2
Re
[
M aˆ(τ, f ) aˆ∗(τ, f )
]
=
∣∣∣S1(τ, f )∣∣∣2 − Ez
σ2
+ K.
(2.7)
The goal of a navigation receiver is to estimate the pseudo-range, calculated from
the code-phase estimates τˆ, of each transmitted signal in order to calculate the position.
The residual frequency, fˆ , is also desired for reasons that will be explained in Chapter 3.
The maximum-likelihood estimates of the synchronisation parameters, τˆ and fˆ , are
obtained by maximising the likelihood function presented in Equation 2.7. Both these
parameters only appear in S1(τ, f ), hence the entire estimation process only requires the
output of the correlators, and the maximum-likelihood synchronisation parameters are
τˆ, fˆ = arg max
τ, f
∣∣∣S1(τ, f )∣∣∣.
We do not explain how the maximisation is performed at this stage, since it is usually
performed at the same time as signal detection in a GNSS receiver. This point will be
treated in more detail in Section 2.3.
We can also express the complex amplitude parameter as a combination of signal
magnitude and residual phase. The magnitude is generally dealt with in the same
way as we estimate the complex amplitude, using a maximum-likelihood estimate.
However, the phase can be treated using a Bayesian approach, as proposed by Kay
(1993). A common solution is to consider a uniform prior distribution for the phase,
which leads to a closed-form expression of the likelihood function. However, the
posterior distribution of the parameters can not, in general, be expressed as a closed-
form expression. Pany (2010) notes that the Bayesian approaches used to estimate the
synchronisation parameters of a GNSS receiver do not bring a significant advantage
over maximum-likelihood methods.
40
2.1. Parameter estimation
Theoretical bounds of the code-phase estimator
The Cramér-Rao lower bound (CRLB) is a lower limit on the variance of an estimator of a
non-random parameter. It is commonly encountered when comparing the performance
of estimators based on the maximum-likelihood approach. The CRLB is a lower limit,
which is not strictly the lowest limit in the general case. It is often possible to derive
“tighter” limits to the performance of an estimator than the CRLB. The posterior
Cramér-Rao bound is the Bayesian equivalent of the CRLB and it can be used to analyse
the performance of single-point Bayesian estimate. I only used the frequentist definition
of the CRLB.
The code phase is used to calculate the pseudo-range, which is then used in the
position equation to determine the user position. Hence, the CRLB of the code-phase
estimator is an important measure of the performance of a navigation system. Pany
(2010) recalls that the CRLB of a code-phase estimator based on the correlation method
is
Var (τˆ) ≥ − 1
MTsR′′c,c∗(0)C/N0
,
where R′′c,c∗(t) is the second derivative of the autocorrelation function of the spreading
code, and C/N0 is the carrier-to-noise ratio, with C the carrier power and N0 the
noise power density. Intuitively, the function R′′c,c∗(t) contains information about the
“sharpness” of the peak of the autocorrelation function and thus the bandwidth of the
navigation signal.
Similarly, Dardari et al. (2009) cite Kay (1993) and give the CRLB of the code-phase
estimator
Var (τˆ) ≥ 1
8MTs pi2 β2 C/N0
, (2.8)
where β is the effective channel bandwidth and is directly related to the channel
frequency response, P( f ), by
β2 =
∫ +∞
−∞ f
2 |P( f )|2 d f∫ +∞
−∞ |P( f )|2 d f
.
Finally, the approach of Weiss et al. (1983) is not often quoted in the literature,
although they propose tighter bounds of the time-delay estimator using a matched
filter. They derive four different bounds, using the Ziv-Zakai method (Ziv et al., 1969),
which are valid over four intervals of the signal-to-noise ratio. They show that, for
signals weaker than a calculated threshold, the code phase can not be recovered and
the estimator is only bounded by the possible parameter space. They also show that the
Ziv-Zakai bound at high signal-to-noise ratios coincides with the Cramér-Rao bound.
41
2. Estimation and detection in navigation receivers
The CRLB is calculated for a simple propagation model, the additive white Gaussian
noise channel without multipath propagation, in all the publications I have found.
We can learn two important points about the CRLB of the code-phase estimator
from these equations. The CRLB becomes smaller, and thus the theoretical performance
of the estimator improves, as the signal strength increases and the signal bandwidth
becomes wider. Wider bandwidths are also privileged in practical applications as this
facilitates multipath mitigation. Intuitively, it is easier for a radio-positioning system to
identify the propagation conditions, and in particular the existence of multiple paths,
when dealing with short pulses. A short pulse over a multipath channel is still visible
if the delays between the paths are longer than the duration of the pulse. Algorithms
can then identify the channel propagation conditions and use this information to
mitigate the effects of multipath propagation. This possibility is a part of the rationale
for ultra-wideband systems, which try to achieve the widest possible bandwidth to
improve the position accuracy in challenging environments.
Estimation in dynamic linear systems
I now address the estimation of parameters in dynamic linear systems. This class
of problems has been studied extensively, in particular since the introduction of the
Kalman filter (Kalman, 1960). I recall only the main operations in a linear Kalman filter
and more information can be found in Brown et al. (1997), for example.
The state of the dynamic system at time step k is represented by its state vector,
xk. The observations, or measurements, zk are related to the state vector by the
measurement function, expressed as a matrix Hk. The linear Kalman filter assumes that
the measurements function is a linear function and the noise is additive white Gaussian
noise. The measurement equation is
zk = Hk xk + Rk ν,
where Rk is the covariance matrix of the observation noise and ν is a vector of
independent normal random variables.
The state vector evolves over time according to linear dynamics, represented by the
transition matrix Fk. This evolution is affected by the process noise, which is modelled as
an additive white Gaussian noise. The process noise represents the noise introduced
during the evolution of the system and, more generally, the uncertainty about the
model. The process model is given by the linear equation
xk = Fk xk−1 + Qk ξ,
42
2.1. Parameter estimation
where Qk is the covariance matrix of the process noise and ξ is a vector of independent
normal random variables.
We now briefly introduce the two main steps of the Kalman filter. More information,
including the prediction and the update equations, is provided in Brown et al. (1997).
The Kalman filter equations provide an estimate of the state vector, represented as a
Gaussian random variable. At the time step k − 1, the prior state is represented by
the probability distribution p(xk|xk−1), which is a Gaussian distribution of mean xˆk|k−1
and covariance matrix Pk|k−1. A measurement, zk, is then taken at time step k and the
update equation is used to calculate the distribution of the posterior state, p(xk|zk). The
state is now represented by a Gaussian random variable of mean xˆk|k−1 and covariance
matrix Pk|k−1. The prediction equation is then applied to the posterior estimate in order
to obtain the prior distribution at the next time step, p(xk+1|xk). The Kalman filter thus
alternates between the prediction and update equation in order to maintain an estimate
of the state vector.
I now introduce the innovation vector and the log-likelihood function of a linear
dynamic model, two notions that will be used in Chapter 4. The innovation, k, is
defined as
k = zk −Hk xˆk|k−1,
and the innovation covariance is
Σk = Hk Pk|k−1 H∗k + Rk.
The normalised innovation squared (NIS), ek, is defined as
ek = k Σ−1k 
T
k . (2.9)
The normalised innovation squared is a useful measure of the goodness of fit of the
model to the observations. It can be shown that, under the noise and dynamics
assumption of the model, ek follows a χ2 distribution with d degrees of freedom,
where d is the dimension of the state vector. The NIS is also used to calculate the
log-likelihood function of the linear dynamic model. The logarithm of the likelihood
function, L(xˆk|z1:M), is
logL(xˆk|z1:M) = −12
M∑
k=1
ek − 12
M∑
k=1
log |Σk| − Nd2 log(2pi) . (2.10)
Ristic et al. (2004), among others, shows how to calculate the CRLB of the state
vector estimate.
43
2. Estimation and detection in navigation receivers
2.2 Signal detection
We now seek to decide whether a signal is present among noisy observations. This
problem is similar to hypothesis testing: either the observations contain both the signal
and the noise, or they contain only the noise. The standard hypothesis testing problem
assumes that the true parameters of the signals are known, under both hypotheses.
We do not have such a knowledge with navigation signals. The code phase and the
frequency residual parameters, for example, are not known since they are dependent on
the user’s position, which is what we want to estimate in the first place. The hypothesis
testing problem with uncertain parameters is called a composite hypothesis problem. I
first recall composite hypothesis testing and I discuss the case of pseudo-noise signals.
Composite hypothesis testing
A binary hypothesis test is composed of two hypotheses, H0 and H1. Each hypothesis
is associated with a probabilistic model, which is described by its likelihood function.
We note L(H0|x) and L(H1|x) the likelihood functions of the model under the H0 and
H1 hypotheses, respectively. It can be shown that the optimal test is based on the ratio
of the likelihood (see, for example, Van Trees, 1968). This leads to the likelihood-ratio
test (LRT), which is
L(H1|x)
L(H0|x) Q η, (2.11)
where η is fixed value called the detection threshold. The test decides for the H1
hypothesis if the likelihood ratio test is greater that η, and for the H0 hypothesis
otherwise.
How the threshold, η, is chosen depends on the detection criterion that is used.
There are two common criteria to choose the threshold from: the Bayes criterion and the
Neyman-Pearson criterion. The Bayes criterion relies on decision theory; it requires that
the prior probabilities of the hypotheses and the costs of making the wrong decisions
are known. The Neyman-Pearson criterion does not require such information and
the performance of the criterion is expressed in terms of probability of false-alarm and
probability of detection. The probability of false alarm is the probability of deciding
H1 whereas H0 is true, and the probability of detection is the probability of correctly
detecting H1. The Neyman-Pearson criterion is more common that the Bayes criterion
in signal analysis, since the prior probabilities of the hypotheses are generally not
known. Berger (1985) provides an analysis of the foundations of decision theory when
applied to statistical problems. For now, I assume that the threshold η is known,
without specifying how it is calculated.
44
2.2. Signal detection
Equation 2.11 assumes that the likelihood functions does not depend on any
parameter. The generalised likelihood-ratio test (GLRT) is often used when the likelihood
functions do depend on parameters, like in composite hypothesis problems. The
GLRT is similar to the likelihood ratio when both likelihood functions are set to their
maximum-likelihood values. Formally, the generalised likelihood ratio is
Λ(x) =
maxθ∈Θ|H1 L(θ|x)
maxη∈Θ|xL(θ|x) ,
and the GLRT is
Λ(x) Q η.
Detection of pseudo-noise signals
We now use the generalised likelihood-ratio test to detect a pseudo-noise signal of
unknown parameters in additive white Gaussian noise. The hypothesis H1 is the
hypothesis that the observations contain both signal and noise, whereas the hypothesis
H0, called the null hypothesis, is the hypothesis that only noise is present. The likelihood
function under the H1 hypothesis is presented in Equation 2.7. The likelihood function
under the null hypothesis is the same as Equation 2.7, with the amplitude parameter
a set to zero. We assumed the noise variance is known and constant, hence Ez(θ)
does not depend on any parameter. The log-likelihood functions under the H0 and H1
hypotheses are, respectively,
logL(θ,H0|z1:M) = K − Ez
σ2
.
and
logL(θ,H1|z1:M) = K +
∣∣∣S1(τ, f )∣∣∣2 − Ez
σ2
.
The GLRT can be expressed using the log-likelihood functions, which gives
log Λ(z1:M) = logL(θˆ,H1|z1:M) − logL(θˆ,H0|z1:M)
=
∣∣∣S1(τˆ, fˆ )∣∣∣2
σ2
.
(2.12)
The GLRT equation, described by Equation 2.12, only depends on the correlator value
and the noise variance. This shows that the output of the correlator, S1(τ, f ), is the only
variable needed both for the estimation and the detection step. These two steps are
generally combined to form the acquisition step.
45
2. Estimation and detection in navigation receivers
2.3 Acquisition of GNSS signals
The goal of the acquisition of a pseudo-noise signal is to estimate the values of the
synchronisation parameters as well as to detect whether a signal is present at the
receiver. It is thus a combination of the estimation and detection steps. The existing
literature often introduces the acquisition process as an operation of its own, however
it may be useful to remember the dual nature of the acquisition process.
Finding the synchronisation parameters
The generalised likelihood-ratio test, shown in Equation 2.12, is commonly used for
the acquisition of GNSS signals. It only depends on the value of the correlator output,∣∣∣S1(τˆ, fˆ )∣∣∣2, which is commonly called the decision variable. We first notice that maximising
the likelihood function of the H1 hypothesis, L(θˆ,H1|z1:M), is similar to maximising the
decision variable of the GLRT. This observation explains why it is easier to perform
both the estimation and the detection in a single step.
So far, we have not discussed how the maximum-likelihood estimates of the
frequency residual and code phase are found. As seen in Section 1.2, the PN code
sequence of a GNSS signal is designed such that the value of its autocorrelation function
is negligible everywhere except near the time origin. The likelihood function has
thus a “sharp” peak around the correct values of the synchronisation parameters, but
is relatively constant elsewhere, as shown in Figure 1.2. The only way to find the
maximum is thus to calculate the likelihood function for all the possible values of the
parameters. This method is in contrast to standard numerical optimisation methods,
which generally use the derivatives of the likelihood function in order to find the
maximum. The likelihood function of PN signals is constant almost everywhere, hence
calculating the derivatives does not help focus the search for the maximum. The set
of all the possible synchronisation parameters for which the likelihood function is
calculated is called the search space. I discuss how the search space is built in Section 2.5.
Figure 2.1 is an illustration of the GLRT applied to the GPS L1 C/A signal. The
figure shows that the correlator output, which is proportional to the log-likelihood
function, has a clear peak around the maximum-likelihood parameters. The function is
negligible elsewhere. The figure also illustrates the effect of the signal bandwidth on
the accuracy of the code-phase estimate, as discussed in Section 2.1. The likelihood
function around the peak is “smoother” for signals with narrower bandwidths, hence
the degradation by the Gaussian noise is relatively more important than for a noise
with same variance but added to a likelihood function with a “sharper” peak.
46
2.3. Acquisition of GNSS signals
21 22 23 24 25 26 27 28 29
Code phase delay (µs)
0
1e+05
2e+05
3e+05
4e+05
5e+05
C
or
re
la
to
r
ou
tp
ut
Figure 2.1: Illustration of the correlator output for different code-phases. The
signal was generated with a true code-phase delay of 25µs. The signal was
representative of a GPS L1 C/A complex baseband signal of carrier-to-noise density
ratio 42 dB−Hz and sampled at 10 MHz. The red curve represents the correlation
value without any filtering, i.e. the bandwidth is limited by the Nyquist frequency.
The blue curve shows the results after a low-pass filter of bandwidth 1 MHz.
Implementation of the detector
We now turn our attention to the decision variable,
∣∣∣S1(τ, f )∣∣∣2. The decision variable is
the squared magnitude of the correlator output. The correlator integrates the incoming
signal with the conjugate of a local replica. The introduction of the magnitude operator
removes the phase mismatch between the received signal and its replica. The phase
removal is also apparent when comparing the log-likelihood from Equation 2.4, before
the complex amplitude parameter is estimated: only the real part of the correlator
output is involved; the magnitude operation is introduced after the residual phase is
estimated in Equation 2.5. The correlator is thus said to perform a coherent integration,
which implies that the phase difference between the received signal, zk, and its replica,
1∗(tk − τˆ) e− j2pi fˆ tk , does not change.
The coherent integration time, Tcoh = M Ts, determines the sensitivity of a receiver,
as shown by Equation 2.15. Intuitively, the longer the coherent integration time, the
more sensitive the receiver is. It is, however, not always possible to extend the coherent
integration indefinitely: any change in the amplitude, frequency or code phase of
received signal during the coherent integration will degrade the performance of the
detector. This can be understood by taking the example of the GPS navigation message.
The navigation message is encoded by a binary phase shifting key modulation, where
the data bits introduce a phase shift of 180°. If the coherent integration is performed
during a bit transition, the integration of the signal after the bit transition cancels the
accumulation before the transition. Similarly, the stability of the local oscillator may
introduce frequency variations and the user motion may affect the signal code phases.
47
2. Estimation and detection in navigation receivers
We can extend the concept of coherent integration to the decision variable. The
sensitivity of the receiver can be improved by non-coherent integration. The non-coherent
detector accumulates the correlator output for Knc non-coherent integrations. This
integration results in the non-coherent detector Y(τ, f ), which is formally defined as
Y(τ, f ) =
1
Knc M σ2z
Knc−1∑
k=0
∣∣∣S(k M+1)(τ, f )∣∣∣2. (2.13)
Y(τ, f ) is now the decision variable of the GLRT detector. The noise variance is
normalised to one by the coefficient before the summation in Equation 2.13. This
normalisation is not required but it facilitates the expression of the probabilities
of detection and false alarm. The acquisition process thus consists of finding the
synchronisation parameters that maximise Y(τ, f ),
τˆ, fˆ = arg max
τ, f
Y(τ, f ).
The decision rule is now
Y(τˆ, fˆ ) Q η,
where η is the decision threshold, calculated for the non-coherent detector. Pany (2010)
offers a more complete discussion about how the non-coherent detector relates to the
maximum-likelihood estimation of parameters when the received signal has a variable
phase.
The GLRT detector needs to calculate the output of the non-coherent detector for all
the possible values of the τ and f parameters in the search space. This two-dimensional
search can be optimised so that the all the values of correlators are calculated in one
operation along the code-phase dimension. We want to calculate the output of the
non-coherent detector, Y(τ, f0) for a discrete set of M candidate code phases, τm = m∆τ
regularly spaced over the search space, at a fixed frequency offset f0. This requires
M evaluation of Equation 2.6,
S1(τm, f0) =
M∑
k=0
zk1∗(tk −m ∆τ)e− j2pi f0 tk . (2.14)
The function shown in Equation 2.14, when considered for the variable m, is a discrete
circular convolution, which can be calculated efficiently using a discrete Fourier
transform. A software receiver can exploit this property to calculate efficiently the M
correlator outputs using a fast Fourier transform (FFT), as explained by Davenport
(1991).
48
2.3. Acquisition of GNSS signals
Theoretical performance
The theoretical performance of the non-coherent detector can be measured in terms
of detection and false-alarm probabilities. Let us consider a signal received with a
carrier power C and a noise power spectral density N0. The carrier-to-noise-density
ratio, C/N0, is often used to evaluate the performance of a detector as a function of
the signal strength. The receiver is similar to the non-coherent detector described in
Equation 2.13, using a detection threshold η, a coherent integration time Tcoh, and Knc
non-coherent integrations. The probability of detection is
pd = QKnc
(√
λ,
√
η
)
, (2.15)
with λ = 2KncTcohC/N0 and Q(a, b) is the Marcum Q-function. λ is sometimes called
the post-integration signal-to-noise ratio. The probability of false alarm is
pfa = 1 − Γ˜Knc
(
η/2
)
, (2.16)
where Γ˜k(x) is the regularised Gamma function. The Marcum Q-function and the
regularised Gamma function are commonly found in scientific software packages.
These two probabilities are used in the Neyman-Pearson criterion in order to choose
the detection threshold, η. There is a trade-off between these two probabilities: the
probability of false-alarm and of detection both increase as the threshold gets lower.
However, the goal is to obtain a high probability of detection and a low false-alarm
rate. This goal can be achieved by either increasing the integration length, reducing the
noise figure of the receiver, or, if practical, improving the signal strength.
Tracking of synchronisation parameters
It is common to track the synchronisation parameters once their initial values have
been calculated, without having to restart the entire acquisition process. The tracking
is generally performed by a phase-locked loop for the residual frequency and by
a delay-locked loop for the code phase. The theoretical analysis and the practical
implementation of tracking have been extensively studied; a reference, among others,
is Parkinson et al. (1996), who investigate the design of tracking loops for GPS receivers.
Most of the work presented in this thesis is focussed on assisted receivers, which do not
require tracking loops. I implemented tracking loops in my software GNSS receiver in
order to decode the navigation message. Nevertheless, we do not need any theoretical
background about tracking loops for the work presented in this thesis.
49
2. Estimation and detection in navigation receivers
2.4 Position estimation
One of the final tasks of a GNSS receiver is to calculate the position by solving the
position equation, presented in Section 1.1. We only consider the TOA position equation
in this thesis, although the TDOA equation can be solved with a similar method. In
practice, Ju-Yong Do et al. (2007) have shown that the TOA approach leads to more
robustness when the statistics of the measurement noise are not fully known.
We now extend the position equation, Equation 1.1, for the case of pseudo-noise
signals. The code phases, τi, of the pseudo-noise signals transmitted from satellite
number i, are directly related to the pseudo-range, ρi. The difference between the two
comes from the uncertainty introduced by the repetition period, TCA, of the PN code
of the GPS C/A code. The pseudo-ranges are calculated by decoding the navigation
message and identifying a pattern of navigation bits that is common to all the signals. In
practice, the time-of-week field is almost always used as it is also required to calculate
the satellite positions at the time of transmission. However, other patterns can be
used if the positions of the satellites are already known. The number of full code
periods, NTOW is then counted from this field and added to the code phase to form the
pseudo-range measurement ρi,
ρi = c (τi + NTOWTCA) . (2.17)
Solution to the position equation
We assume the code-phase estimation step previously described has calculated the
pseudo-range estimates, ρˆi, and the uncertainty is represented by additive white
Gaussian measurement noise. The TOA position equation with N noisy pseudo-range
measurements can thus be written
ρˆi = ||xˆ − xi|| − cbˆu + νi,
where ν = [ν1, . . . , νN] is a multivariate Gaussian random vector of covariance matrix
Σρ, representing the measurement noise.
The position equation is non-linear and an approximate solution of the maximum-
likelihood position estimate is usually found using the iterative least-squares algorithms,
an approach which is found in early receivers (Parkinson et al., 1996, chapter 9), modern
software receivers (Borre et al., 2007), and assisted receivers (van Diggelen, 2009). The
basis of the method is to linearise the position equation around a coarse estimate,
solve the equation using linear methods, and iterate with the new estimate. The first
50
2.4. Position estimation
derivatives, which are required to linearise the function, are
ui =
δρi
δx
=
xˆ − xi
||x − xi||
and
δρi
δbˆu
= −c
ui is also the unit vector between the user position and the satellite i. The unit vectors
are often used to form the geometry matrix, G, commonly encountered in positioning
problems. G is defined as
G =

u0 −c
...
...
uN −c

We introduce the augmented state vector, θ = [x, . . . , bu], which contains all the
parameters we wish to estimate. The first-order linearisation of the position equation
leads to the difference equation
∆ρ = G ∆θ + , (2.18)
where  is the approximation error and ∆ρ =
[
∆ρ1, . . . ,∆ρN
]
is the difference between
the theoretical and observed pseudo-range vector.
The maximum-likelihood estimate, θˆ, is approximated by an iterative algorithm.
An approximate position, generally the previously calculated position, is used as an
initial value of the estimate, θˆ(0). The following algorithm is then iterated over k:
∆ρ(k)i =
∣∣∣∣∣∣xˆ(k) − xi∣∣∣∣∣∣ − c bˆu(k) for each i
∆θ(k) =(Gt Σ−1ρ G)−1 Gt R−1 ∆ρ(k)
θˆ(k+1) =θˆ(k) + ∆θ(k)
where xˆ(k) and bˆu(k) are the estimates at iteration k, contained respectively in the first
three components and the last component of θˆ(k).
The vector ∆ρ is called the vector of pseudo-range residuals and ∆θ the vector of
parameter residuals. The residuals are often used to check the behaviour of the
algorithms. It can be shown, for example, that the distribution of the residuals follows
a Gaussian distribution, under the AWGN noise assumptions.
There are different possible stopping rules for the iteration loop. A common choice
is to stop when the sum of the squares of the residuals is below a given threshold.
The position equation is generally a well-posed problem and the algorithm generally
converges in less than ten iterations..
51
2. Estimation and detection in navigation receivers
Dilution of precision
We can calculate the Cramér-Rao bound of the position estimate calculated by the
iterative least-square algorithm. With Σρ the measurement noise covariance matrix of
the measurement noise and G the geometry matrix, the CRLB matrix is
CRLB−1 = G Σ−1ρ GT. (2.19)
The matrix Σρ is generally not known accurately. The noise variances of the pseudo-
range measurements can be estimated from Equation 2.8, which depends on the signal
strengths. However, that equation is only a lower bound, and effects such as multipath
propagation and the non-Gaussianity of the noise affect the exact noise variance. The
cross-correlations between the pseudo-range measurements of the different satellites
are generally unknown, although the propagation conditions around the user may
introduce correlations. As a result, the measurement noise covariance, Σρ, is generally
unknown. It is often convenient to analyse the theoretical performance of a receiver
based on the transmitter geometry alone. These effects are summarised in the dilution of
precision (DOP) metric, which isolates the effects of the measurements noise by setting
the noise covariance matrix to the identity matrix. The DOP matrix, DOP, is thus
DOP = G GT. (2.20)
Specific measures of the DOP are often encountered. The horizontal dilution of precision
(HDOP) measures the effect of the geometry on the horizontal position only. It is equal
to the trace of the horizontal components of the DOP matrix. Chaffee et al. (1994) give a
more complete treatment of the theoretical limits of the position calculation.
Solution in the position domain
The iterative least-squares is the most common method to calculate the user position.
Alternative methods have been proposed recently, which are based on the direct
calculation on the maximum-likelihood estimate of the position, without the interme-
diate step of estimating the synchronisation parameters. Closas et al. (2007) recall the
likelihood function in the position domain and propose a numerical approach to find
its maximum. Axelrad et al. (2011) investigate the advantages of the direct positioning
method with weak signals. An advantage of methods based on the likelihood function
is that it combines the signals from all the satellites, even if the signal strengths of some
individual satellites are too weak to be able to estimate the synchronisation parameters
– the information contained in these weak signals is still included in the solution. One
52
2.5. Assisted-GNSS receivers
of the problem faced with this method is the size of the search space, which is now
composed of the three spatial dimensions plus the user bias. A good estimate of the
user position, and possibly the user clock bias, is required in order to reduce the search
space to a tractable size. The need for a fourth variable in the search, for the user bias,
is not always explicitly mentioned in the literature. I considered the direct positioning
approach in my work, in particular because the method could benefit significantly from
an estimate of the user clock bias, however the results of my early investigations are
not reported in this thesis.
2.5 Assisted-GNSS receivers
I now describe the operation of assisted-GNSS receivers, typically encountered in
cellular devices. The literature about assisted receivers is not as large as the one about
conventional receivers, despite the wide popularity of assisted receivers in consumer
devices. An analysis of the architecture and the performance of these receivers is
provided by van Diggelen (2009).
Description
An assisted-GNSS receiver is a type of receiver that uses external information, called
assistance data, in order to improve the positioning and navigation performance. The
assistance data generally consists of the approximate universal time and position of the
receiver, as well as the ephemeris data of the GNSS constellation.
Tracking loops, which are used to decode the GPS navigation message, are generally
absent in an assisted-GPS receiver since the navigation message is provided by the
assistance data. However, decoding the TOW field from the navigation message is
required in order to calculate the pseudo-range, as shown by Equation 2.17. The only
information available to an assisted receiver in order to calculate the position is the
code phase, which differs from the pseudo-range by an integer number of code periods.
We can still calculate the user’s position with only code phases, by using the position
equation
τi =
1
c
||xi(TOTi) − x|| − bu + ni TPN, (2.21)
where ni is the integer number of repetitions of the PN code since the time at which the
signals have been transmitted and TOTi is the unknown time of transmission of the
signal coming from satellite i. The satellite positions, xi (TOTi), are now dependent on an
unknown variable whereas a conventional receiver can calculate the satellite positions
53
2. Estimation and detection in navigation receivers
from the known times of transmission of the signals. A conventional GPS L1 C/A
receiver, for example, can decode the TOW field of the navigation message and calculate
the satellite positions at that epoch, using the ephemeris carried in the navigation
message. An assisted receiver, on the other hand, solves the position equation by
adding TOTi to the set of unknown variables, which now becomes (x, bu,TOTi). The
integer number of code periods, ni, is unknown but is not part of the set of unknown
variables. It can be calculated directly if the approximate universal time and position
of the receiver are known within some maximum error bounds. Similarly, TOTi is
constrained to be an integer number of code periods, hence all the times are known
as soon as the TOT of a specific i has been calculated. These methods are often called
millisecond-ambiguity positioning methods, in reference to the 1 ms code period of the
C/A code. The measurements are now the fractional pseudo-range, cρi, in contrast to
the full pseudo-range measurements given in Equation 2.17.
Lannelongue et al. (1998) and van Diggelen (2009) show how the position can be
calculated with a least-squares algorithm similar to the one presented in Section 2.4.
There are now five unknown variables, thus requiring the pseudo-range measurements
from at least five satellites in order to solve the position equation. Sirola et al. (2002)
show how the position equation converges to the correct solution if the initial time
and position are known within some error domain. The error domain that guarantees
the convergence has a conical shape, which extends over the maximum time errors of
±3.5 min and a diameter which is the maximum position error of 150 km for the GPS
C/A code.
Assisted-GNSS receivers have multiple advantages that make them desirable in
consumer devices. The absence of tracking loops simplifies the design of the receiver
and reduces the physical size of the chip, which facilitates its integration on existing
semiconductor chips. On the one hand, not needing to decode the navigation message
has the drawback of requiring one more unknown variable than in a conventional
GPS receiver. On the other hand, it has the advantage of allowing the operation of
the receiver with weaker signals, since the signal strength required to decode the
navigation message is generally higher than the signal strength required to estimate the
pseudo-ranges. Furthermore, the time-to-first-fix (TTFF) is improved in assisted-GNSS
receiver, using two mechanisms. The receiver does not need to decode the satellite
ephemeris data, which can take up to 30 s using the GPS L1 C/A signal, since the
ephemeris is contained in the assistance data. The millisecond-ambiguity positioning
also improves the TTFF since the receiver does not have to identify and record the
TOW field. For these reasons, some receiver manufacturers also use the term snapshot
54
2.5. Assisted-GNSS receivers
or push-to-fix positioning to describe this approach.
Assistance data
The performance of an assisted receiver is dependent on the amount and quality of the
assistance information available. We only consider the case of assistance data provided
by a cellular communication networks, for example a GSM network. Assistance data is
not always available and, when available, the level of information is contingent on the
assistance technologies used by the cellular network operator. Position assistance is
provided by cellular positioning technologies such as the ones reviewed in Section 1.3.
The time assistance is dependent on the cellular network architecture. Unsynchronised
networks, such as GSM networks, are generally less capable of providing accurate time
information than are synchronised networks. The time accuracy achieved in a GSM
network is the topic of Chapter 4.
Cellular transmitters usually employ high-quality oscillators to generate the radio-
frequency carriers. The accuracy of the carrier frequency can be used to correct the
frequency bias of the local oscillator of the receiver. This form of assistance is not
directly explicitly provided by the cellular network to the acquisition engine of the
GNSS receiver. Instead, it is implemented in the handset transceiver, which locks
its local oscillator onto the radio-frequency carrier. The same local oscillator can be
used by the GNSS receiver, hence the stability of the local oscillator is the same as
the stability of the oscillators in the cellular BTSs. The apparent carrier frequency is
affected by the Doppler effect caused by the velocity of the mobile device and by the
frequency variations caused by multipath propagation. The stability of an oscillator
locked onto the carrier frequency is therefore not necessarily as good as the stability
of the BTS clocks. Furthermore, the frequency accuracy and stability provided by the
cellular network does not guarantee the time accuracy or the phase stability of the GSM
synchronisation signals, as the RF carrier and the cellular signals may be controlled by
different oscillators.
There is no general agreement as to what typical levels of assistance should be
considered and terms such as “coarse assistance” and “fine assistance” have different
meanings for different authors. I consider three different scenarios all along this thesis,
outlined in Table 2.1. The low-level scenario is typical of a mobile device without
cellular connectivity. The time is provided by the internal clock, the position is taken
as the previous calculated position, and the short-term ephemeris is not available.
The medium-level scenario corresponds to a device operating in an unsynchronised
network, for example a GSM network. Cell-ID is generally available, although its
55
2. Estimation and detection in navigation receivers
position uncertainty is similar to the radius of a cell. The local oscillator is locked on the
RF frequency, yet the timing signals of the unsynchronised network can not provide
time accuracy better than a few seconds of the universal time. Finally, the high-level
scenario is representative of a synchronised cellular network, which can provide the
universal time with a good accuracy. The time uncertainty mainly comes from the
uncertainty in the time of flight of the signal from the base station.
Table 2.1: Scenario assumptions.
Aiding Data Errors Oscillator
scenario network Position Time stability Almanac Ephemeris
Low level No ≤ 300 km ≤ 5 min 1 ppm Yes No
Medium level Yes ≤ 30 km ≤ 5 s 0.1 ppm Yes Yes
High level Yes ≤ 30 km ≤ 10µs 0.1 ppm Yes Yes
Assisted receivers also differ in how the position is calculated. The synchronisation
parameters are generally estimated by the mobile device, but the position calculation
can be performed using either an MS-based or MS-assisted method. The position is
calculated on the device in an MS-based method. On the other hand, an MS-assisted
method consists of transmitting the synchronisation parameter estimates to a location
server on the cellular network; the server calculates and then returns the position to the
mobile station. The MS-assisted method has the advantage of simplifying the design
of the mobile device and it may also exploit more information than is available on
the device. The main drawback is the need for a data access to the cellular network.
An even more decentralised method has been proposed in the literature, in which the
baseband signal is simply transmitted to a server that calculates the position.
2.6 Summary
In this chapter, we have reviewed the bases of the acquisition of direct sequence-
spread spectrum signals, which are often encountered in GNSS and cellular systems.
I highlighted the distinction between parameter estimation and signal detection,
although the acquisition process may also be understood using different approaches,
such as by introducing the matched filter. I outlined the foundations of frequentist and
Bayesian estimation. We looked at the estimation of the synchronisation parameters
of pseudo-noise signals and the theoretical bound on the code-phase estimation. We
also investigated the estimation of the user position by using an iterative least-squares
algorithms and we introduced the dilution of precision as a theoretical bound of the
56
2.6. Summary
position error. Finally, we reviewed the operation and architecture of assisted-GNSS
receivers. These results will be continuously used in the remainder of this thesis.
57

CHAPTER 3
Analysis of the time-to-first-fix in
assisted-GNSS receivers
The time-to-first-fix (TTFF) is a metric commonly used to evaluate the performance of a
GNSS receiver. It is informally defined as the time taken by the receiver to calculate
a position, also called a fix, from the instant the user has turned on the receiver. This
metric is particularly important for location-based services on mass-market devices,
such as smartphones, for which a quick position fix is often more valuable than an
accurate position. Many applications can work with an approximate position; however
the response time of the application, which is strongly dependent on the TTFF, must
remain short to ensure a positive user experience.
Optimisation of the TTFF is a challenging problem to the GNSS receiver designer.
The current options are either to run Monte-Carlo simulations with a theoretical receiver
model or to repeat several acquisitions on a real receiver connected to a signal simulator.
Both these techniques are time-consuming, which hinders the optimisation of the
TTFF. An alternative approach is to develop an analytical model that can be analysed
and optimised more easily. This approach has been under investigation since the
introduction of spread-spectrum signals, with Holmes et al. (1977) and Hopkins (1977)
presenting early studies.
Analytical methods have been limited to calculating statistics, such as the mean
and the standard deviation, of the acquisition time of just one satellite. Extracting
information about the TTFF from such descriptive statistics, the mean time to acquire all
the satellites for example, cannot be calculated from the mean times of each acquisition
individually. Extending these methods, which are valid for the acquisition of a single
59
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
satellite, to the case of multiple satellites is not straightforward. Additional information
is needed, such as the probability distributions of the individual acquisition times.
Typical GNSS receivers require at least 4 satellites in order to calculate a position, so
that existing methods cannot be used directly to analyse the TTFF. It may also be
desirable to calculate the performance at a specific confidence level, for example the 90%
probability of the TTFF. This is not possible if the probability distribution is unknown.
In this chapter, I derive the theoretical distribution of the TTFF of an assisted-GNSS
receiver. First, I review previous studies on the acquisition time of spread-spectrum
signals and highlight some properties of probability generating functions (PGF). I then
investigate the model of an assisted receiver and I discuss the assumptions made. I
then calculate the probability distribution function of the TTFF using this model and
results of previous studies. I illustrate the method with a simple scenario, and provide
further analysis by using real measurements. Finally, I discuss the limitations and the
opportunities for improving this new method.
An early version of this work was presented in Couronneau et al. (2011), where the
acquisition of one signal only was considered. I extend that work in this chapter by
modelling the two-stage acquisition process of an assisted receiver.
3.1 Analysis of the acquisition of direct-sequence spread
spectrum signals
The theoretical acquisition time of pseudo-noise signals has been investigated since
the introduction of spread-spectrum systems and CDMA communications. Different
techniques have been proposed and, in this chapter, we will focus on the most general
method and its application to GNSS receivers.
Signal acquisition model
The goal of the acquisition process is to find the most likely set of synchronisation
parameters of the signal. In spread-spectrum communications, the synchronisation
parameters are the parameters that a receiver needs in order to decode the received
signal. The synchronisation parameters for the signal transmitted from satellite k are the
PN code phase, τk, and the frequency offset, ∆ f k. These can be represented by the pair
θk =
(
τk,∆ f k
)
. The PN code sequence, ck(t), is sometimes considered a synchronisation
parameter, although I do not make this distinction since the PN code sequences of the
GPS satellites in view of the users are generally known.
60
3.1. Analysis of the acquisition of direct-sequence spread spectrum signals
The set of all the possible parameters is commonly named the search space. Its size
and content depend on prior information available about the positions, velocities, and
time offsets of each transmitter and the receiver. More about the size of the search space
is discussed in Section 3.5.
The PN code sequences are almost always searched in parallel, in Nch acquisition
units called channels. This stems from the need to acquire at least four satellites at
the same time in order to obtain a position fix. In general, the number of PN code
sequences need not be the same as the number of satellites, since satellites increasingly
transmit more than one GNSS signal. Here, however, I will consider that every code
sequence belongs to a different satellite. Bastide et al. (2001), among others, treat the
acquisition of multiple GNSS signals from one satellite.
Subsets of the search space of the synchronisation parameters may also be processed
in parallel. This can be achieved via hardware techniques, such as massively parallel
correlators (see van Diggelen, 2002), and software techniques, such as FFT-based
correlation seen in Section 2.3. However, it is currently not practical to carry out a
search over the entire search space in one operation, so Nw subsets, Θi, of the search
space are employed. These subsets are often called tiles or windows in the spread-
spectrum literature. We name the window which contains the correct parameters an
“H1 window”; other windows are labeled “H0 windows”. Here, the model allows for
only one H1 window, a simplifying assumption, which I will discuss at the end of this
chapter in Section 3.6.
The time taken to process a window is called the dwell time and hence sets the
granularity in time of the search.
The synchronisation parameters are generally estimated via the generalised likeli-
hood-ratio test, introduced in Section 2.3. It consists in calculating the likelihood ratio,
or equivalently the output of a matched filter, for all candidate parameters within
a window, and retaining the parameters that produced the largest output. If the
likelihood ratio crosses a pre-determined threshold β, a detector hit is declared.
The resulting search procedure is sometimes called a hybrid search. First, the signal
parameters are searched within a window, one dwell time at a time. If a hit occurs,
the search terminates with the synchronisation parameter estimates θˆk. Otherwise,
the search continues onto the next window. If no hit has occurred after all windows
have been explored, the whole process is repeated. A hit simply means that the
likelihood-ratio of the parameter combination θˆk is the largest in its subset and has
crossed the detection threshold: it could be a false-alarm.
Sequential acquisition schemes, also referred to as variable dwell time, have been
61
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
studied as well as serial search strategies. These schemes try to minimize the acquisition
time by performing a near continuous signal detection with a time-decreasing threshold.
Some of the first results were published by Gumacos (1963), who shows how the
sequential probability ratio test (SPRT) leads to the optimal acquisition time. More
recently, sequential detection has been investigated for GNSS signal acquisition by
Yong-Hwan Lee et al. (2000). The theoretical analysis of sequential acquisition is
particularly difficult. O’Mahony et al. (2011) have recently provided some theoretical
results of the sequential acquisition of GNSS signals.
In this chapter, I consider only architectures with a single, fixed, dwell time.
Theoretical analysis of acquisition time
The theoretical acquisition time of DSSS signals has been studied for many decades.
Holmes et al. (1977) represented the acquisition process as the flow graph of a Markov
chain and they were able to calculate the mean acquisition time in a single dwell-time
and a double dwell-time sytem. Hopkins (1977) also represented the process as a
Markov chain and derived similar results, for the case of a single dwell-time detector,
by directly solving the Markov chain transition equations. Finally, Polydoros et al.
(1984a,b) reviewed and unified the methods available at that date and introduced a
framework often known as the flow-graph method. More recently, Lohan et al. (2004)
and O’Driscoll (2007) applied flow-graph methods to the acquisition of GNSS signals.
They also mention for the first time that the discrete Fourier transform can be used to
calculate the full probability distributions of the acquisition time variables.
The flow graph of the search process described previously is represented in Figure 3.1.
The nodes represent the different states of the acquisition process. There are Nw − 1
nodes of type H0 representing the H0 windows and one H1 node representing the H1
window. The shaded nodes are absorbing states, i.e. they are the possible final states
of the search. There are three absorbing states: the D state, corresponding to a true
detection; the FA|H0 state, corresponding to a false-alarm in an H0 window; and the
FA|H1 state, corresponding to a false-alarm in the H1 window.
The arcs represent the possible transitions between the nodes. The transition
probabilities are related to the detection probabilities, calculated at the window-level:
PFA|H0 is the false-alarm probability in an H0-window; PFA|H1 in an H1-window; PMD
the probability of missed detection; and PD the probability of detection. It is known
how to calculate these probabilities and the results will be recalled in Section 3.2.
The search can start in any window i, with probability pii. This is represented on the
flow graph by the incoming arcs to the H0 and H1 nodes. It may be possible to focus
62
3.1. Analysis of the acquisition of direct-sequence spread spectrum signals
1 − PFA|H0
H0 H0
PFA|H0
PMD
PD
PFA|H1
H1
D
FA|H1
FA|H0
pi1
H0
pi2
piNch
Figure 3.1: Flow graph of a single acquisition. The circles, or graph nodes,
represent the different states of the acquisition process. The lines between the
nodes are the transition, whose probability of occurence is written above the line.
The shaded circles represent the possible final states of the acquisition. The search
can start in any of the H0 and H1 with probability pii.
the search in the most likely windows, depending on the information available about
the location of the synchronisation parameters. For example, the expanding window
and the Z-search algorithms implement this strategy. In general, the search can start in
any window with a uniform probability, so pii = 1Nw , and I will make this assumption in
the remainder of this work.
Probability generating functions
At the core of the flow-graph method is the notion of a probability generating function
(PGF). The probability generating function of a discrete random variable fully charac-
terizes the distribution of the random variable as a power series. A discrete random
variable, X, taking values in {0, 1, . . . ,M} with probability mass function P(X = m), has
63
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
a probability generating function defined by
UX(z) =
∞∑
k=0
zk P(X = m),
with z ∈ C. The definition domain C is a region of the complex plane Cwhere the series
converges.
An interesting property is the relationship between the PGF of two discrete random
variables and the PGF of their sum. If X and Y are two discrete random variables of
probability generating functions UX(z) and UY(z) with the same convergence region,
the probability generating function of their sum is
UX+Y(z) = UX(z) UY(z). (3.1)
This additivity property has enabled the probability generating function to become a
common tool in the analysis of branching processes, arrival times, etc.
Although the probability generating function fully describes the distribution of
a random variable, it is not the most convenient or intuitive way to study such a
distribution. The probability mass function (PMF) or the central moments are more
convenient forms of representation. The moments of the distributions, and hence the
means and standard deviations, can be easily calculated from the PGF. In particular,
it can be shown (see, for example, Riley et al., 2006) that the expectation of X can be
calculated from the derivative of the PGF evaluated at z = 1, if it exists, using the
relation
〈X〉 = dUX(z)
dz
∣∣∣∣∣
z=1
, (3.2)
where 〈.〉 is the expectation operator. If z = 1 is a pole of dUX(z)dz , the limit can be used.
Similarly, higher-order derivatives of the PGF are related to the higher moments of the
distribution.
Flow-graph model of single acquisition
The goal of the analysis is to calculate the distribution of the time to reach each of the
possible final states of the acquisition. Holmes et al. (1977) and Polydoros et al. (1984a)
showed how the probability generating functions of these times can be easily calculated
from the flow graph of the acquisition process. We represent the time to reach the final
states, for the nth channel, by the random variables:
• T(n)D , the number of dwell times to reach the D state;
• T(n)FA, the number of dwell times to reach either false-alarm states - FA|H0 or FA|H1;
64
3.1. Analysis of the acquisition of direct-sequence spread spectrum signals
Hr0(z)
H0
H0
H0HFA|H0(z)
HMD(z)
HD(z)
HFA|H1(z)
H1
D
FA|H1
FA|H0
Figure 3.2: Flow graph of the acquisition process with probability generating func-
tions. The notations are the same as Figure 3.1, where the transition probabilities
have been replaced by their probability generating functions. Not pictured on the
diagram, the possible entry point is any of the H0 or H1 windows with uniform
probability 1Nch .
• T(n)hit , the number of dwell times to reach any of the absorbing states, often called
the time-to-first-hit (TTFH).
The flow graph of the hybrid search, expressed using the generating functions, is
shown in Figure 3.2, where
HD(z) = PD z,
Hr0(z) = (1 − PFA|H0) z,
HMD(z) = (1 − PD − PFA|H1) z,
Hl(z) = HMD(z) Hr0(z)
Nw−1,
HFA|H0(z) = PFA|H0 z, and
HFA|H1(z) = PFA|H1 z.
65
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
These generating functions have been calculated by O’Driscoll (2007) for the hybrid
search model and are reproduced here:
UD(z) =
1
Nw
HD(z)
1 −HMD(z)Hr0(z)Nw−1
1 −Hr0(z)Nw
1 −Hr0(z) ,
UFA|H0(z) =
1
Nw
[
(Nw − 1)HFA|H0(z)
1 −Hr0(z) −
[Hr0(z) −HMD(z)] HFA|H0(z)
[1 −Hr0(z)]2
1 −Hr0(z)Nw−1
1 −Hl(z)
]
,
UFA|H1(z) =
HFA|H1(z)
Nw[1 −Hr0(z)]
1 −Hr0(z)Nw
1 −Hl(z) ,
UFA(z) = UFA|H0(z) + UFA|H1(z), and
Uhit(z) = UFA(z) + UD(z).
(3.3)
Inversion of probability generating functions
We could calculate the probability generating functions of T(n)D , T
(n)
FA, and T
(n)
hit using the
flow graph of Figure 3.1. Equations 3.1 and 3.2 could then be used to calculate the mean
and the standard deviation of the acquisition time. This is the approach adopted by,
for example, Polydoros et al. (1984a). Instead, we wish to characterize the distribution
in terms of its probability mass function or its cumulative distribution function. The
relationship between the probability mass function and the probability generating
function is
P(X = m) =
1
2pi j
∮
C
UX(z) zm−1 dz, (3.4)
where the PMF is defined over m = 0, . . . ,M and C is the integration contour in the
complex plane, which must include all the poles of UX(z).
The integral in Equation 3.4 does not admit an analytic solution in general. Cavers
(1978) showed, using the trapezoidal method, that a numerical approximation of the
integral in Equation 3.4 is proportional to the inverse discrete Fourier transform (DFT),
when the integration contour is a circle. In the special case where the integration is
performed on the unit circle of the complex plane, the trapezoidal integration is the
classical inverse DFT. A similar relationship, called the Poisson-Lattice algorithm,
is given by Abate et al. (1992), without explicitly relating it to the DFT. Polydoros
et al. (1984a) have also noted the relationship between the DFT and the probability
generating function in their analysis of acquisition time. Holmes et al. (2006) used a
similar method, proposed by Nuttall (1970).
I now illustrate how the method is applied. We first evaluate UX (z) at L discrete
points of the unit circle zl = e2pi jl/L, to form the vector u = (u0, . . . ,uL−1), where
66
3.1. Analysis of the acquisition of direct-sequence spread spectrum signals
ul = UX (zl). L affects the accuracy of the numerical integral and we choose LM. The
probability mass function is then approximated by
P(X = m) ≈ 1
L
L−1∑
l=0
ul e2pilm/L, (3.5)
for k ∈ {0, . . . ,K}. Equation 3.5 is simply the expression for the m-th coefficient of the
inverse DFT of the vector u. This relationship is also known as the Poisson sum.
Equation 3.5 is used to calculate the truncated probability mass function efficiently.
First, the vector u is calculated. An L-point inverse DFT is then performed over the
entire vector and the result gives an approximation of the PMF truncated to its first L
coefficients. Since most of the probability mass of the acquisition time distribution is
concentrated over short acquisition times and then becomes negligible, the truncated
distribution is a good approximation of the full one.
A naive calculation of the entire PMF from Equation 3.5 would require ML evalua-
tions of the PGF. By comparison, the DFT requires only L evaluations of the PGF, which
are stored in the vector u. Furthermore, the inverse DFT can be performed efficiently
by implementing the inverse Fast Fourier Transform (FFT) algorithm, which is widely
available in scientific software packages.
Another useful property is the relationship between the PGF of the complementary
cumulative distribution function and of the probability mass function. The complemen-
tary cumulative distribution function of a random variable X is the function P(X > x),
which is sometimes called the tail function. We note GX(z) the generating function of
the complementary cumulative distribution. GX(z) can be calculated directly from the
generating function of the probability mass function, UX(z), using the relationship
GX(z) =
z −UX(z)
z − 1 . (3.6)
This relationship is also mentioned by Cavers (1978), among others. Equation 3.6 can be
calculated for the pole z = 1 using the l’Hopital rule, which results in GX(z = 1) = U′X(1).
Equation 3.2 shows that U′X(1) is the expectation of X, which represents the mean
acquisition time in signal acquisition problems. This time has been calculated by several
authors, such as Holmes et al. (1977), and their results can be easily re-used to provide
an analytical expression of U′X(1). Equation 3.6 will often be used in the remainder of
this chapter to calculate the cumulative distribution P(X ≤ k).
Summary of numerical calculation of inversion functions
The conversion of generating functions into distribution functions and the reciprocal
transformation are the essential operations needed to build a theoretical model of an
67
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
assisted receiver. We now summarise the results from the previous paragraph and
highlight the practical aspects of the numerical calculations.
The first step in finding the distribution function P(X ≤ m) from the generating
function UX(z) is to calculate the generating function of its complementary cumulative
distribution GX(z), as shown by Equation 3.6. Then we form the vector g = (10, . . . , 1L−1)
from the probability generating function GX(z) evaluated at L uniformly spaced point
of the unit circle: 1l = GX
(
e2pi jl/L
)
. Finally, we apply the inverse FFT and transform the
result to a distribution function using
P(X ≤ m) = 1 − IFFT (g) (m).
Conversely, one can calculate L points of the generating function over the unit circle,
1l = GX
(
e2pil/L
)
of the random variable X described by its cumulative distribution vector
c, with ck = 1 − P(X ≤ k) using the relation
1l = FFT (c) (l).
Direct derivation of acquisition distribution
DiCarlo et al. (1980) and Meyr et al. (1983) have derived an expression for the distribution
function of the acquisition time of the signal from one satellite. The derivation is based
on a model of the acquisition process expressed as a geometric series, whose geometric
coefficient is the probability of detection PD. This approach can also be used to calculate
the distribution of the decoding time of the navigation messages, as shown by Holmes
et al. (2006), who calculated the decoding time of the GPS P(Y) navigation message,
and Anghileri et al. (2010) who investigated the acquisition time and frame decoding
time of the Galileo signals. However, this approach is not as flexible as the inversion of
the flow-graph generating function, despite being simpler. Besides, as it will be seen
in a later section, the probability generating function is needed to model an assisted
receiver.
3.2 Mean features of assisted-receivers models
In this section, I present the model assumptions, and in particular the specificities of
assisted receivers. The goal is to identify the variables required to build a flow-graph
model of an assisted receiver.
68
3.2. Mean features of assisted-receivers models
Signal search strategy
Manufacturers share little information about the architectures of their receivers. The
signal search strategy is usually unknown and may be a combination of classical acqui-
sition techniques with ad-hoc solutions. Here, we assume that the signal acquisition
unit of the receiver performs a generalized likelihood ratio test, seen in Section 3.1.
The unknown frequency offset of the received signal is a combination of the Doppler
offset, caused be the relative motion of the satellite and the receivers, and by the
frequency bias of the local oscillator. The expected Doppler offset of the satellite is
known with a good accuracy, if assistance data is available – at least for a static receiver.
Hence, after detection of a first satellite, it is possible to estimate the frequency bias of
the local oscillator and remove its contribution to the frequency offset search space. The
size of the initial search space is therefore reduced for the acquisition of the subsequent
satellites.
The acquisition process in an assisted receiver is thus a two-stage process, with two
different search spaces. We will use the term full space to refer to the original search
space, and reduced space to refer to the search space where the contribution of the local
oscillator bias has been removed. It appears that such a two-stage architecture, where
the search space sizes varies between the stages, has not been investigated previously.
We also make the assumption that the receiver has more channels available to it than
the number of satellites in view, an assumption which generally holds for GPS L1 C/A
receivers designed after 1990. We also assume that the detection is “single-dwell”,
meaning that only one threshold crossing is required. The flow-graph method can
easily include double-dwell detectors if required.
Receivers without tracking loops
The absence of tracking loops in assisted-GNSS receivers has two consequences. First,
the time of transmission of the signals, which is usually decoded from the navigation
message, is unknown. We have seen, in Section 2.5, that this problem is solved by
adding one more unknown variable to the position equation than in conventional
receivers. I therefore assumed a minimum of KFIX = 5 satellites were needed to calculate
a three-dimensional position. In Chapter 5, the minimum number of satellites required
for positioning will be reduced by considering fine-time aiding and 2D positioning.
A second difficulty concerns the treatment of false alarms in assisted receivers.
Conventional receivers decode the navigation message and it is then possible to check
the parity field of the navigation message to detect a false alarm, with almost certainty
69
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
(see Parkinson et al., 1996). On the other hand, assisted receivers generally do not decode
the navigation message. Assisted receivers are designed for difficult environments,
where the signal strength may be too weak to decode the data bits reliably. Furthermore,
tracking loops (DLL and PLL) increase the complexity and power consumption of the
receiver. Mass-market receivers sometimes do not include tracking loops at all and are
therefore not able to decode the navigation message.
Assisted receivers must also detect false alarms reliably. To date, it seems that
no analysis of the false-alarm rate of assisted receiver has been done and that most
implementations rely on ad-hoc or heuristic methods. I was not able to find information
about the treatment of false alarm in commercial receivers. Nevertheless, I assume
that the detection of a false alarm is possible. One can imagine a method based on
the residuals of the iterative least-squares algorithm used to calculate the position to
justify that detecting a false alarm in assisted receivers is indeed possible. This least-
squares positioning algorithm assumes the noise of each pseudo-range measurement
is Gaussian, hence the residuals must follow a central-χ2 distribution, with as many
degrees of freedom as there are pseudo-range measurements. It may become possible
to perform a statistical test on the residuals and detect a possible false-alarm if the
position equation is over-determined.
I modeled a false alarm by introducing a fixed penalty time, Tpenalty, given in
numbers of dwell times, in the flow graph.
Signal model
We simplify the analysis by assuming the signal strength is constant over an entire
acquisition period. A more accurate model would be a slow-fading channel, where the
signal power is time-correlated. However, I was not able to include such a model in
closed form in the analysis framework. Abdi et al. (2000), have derived analytic forms
of some second-order statistics for different types of propagation channels. Fast-fading
channels, in which the signal strength fluctuations are much shorter than the acquisition
time, can be modelled by calculating the detection statistics (such as probability of
detection and false alarms) using a parametric model of the channel. See, for example,
Simon et al. (1998).
Detection parameters
The cell-level detector is based on a matched filter, consisting of Knc non-coherent
integrations of length Tcoh, followed by a likelihood-ratio test with threshold β. The
70
3.2. Mean features of assisted-receivers models
received signal strength is C/N0 (in normalized gain units – not dB-Hz). The dwell time
is thus
Tdwell = KncTcoh. (3.7)
In the Neyman-Pearson framework, the detection threshold is chosen to minimize
the false alarm rate and maximize the probability of detection, at a given signal-to-noise
ratio. The statistics of the non-coherent detector for hybrid architecture are given by
Corazza (1996) and Borio et al. (2008b). The main results are recalled here.
Cell level
The equations presented in Section 2.3 are used in order to calculate the following
detector statistics at the cell level: pFA(β) the probability of false alarm and pD(β,C/N0)
the probability of detection.
Window level
The window-level detection probabilities, which are those of the generalized likelihood
ratio test, have been calculated by Borio et al. (2008b) and are recalled here:
PFA|H0 = 1 − (1 − pFA(β))Nc (3.8)
PMD = pD(β,C/N0) pFA(β)Nc−1 (3.9)
PD =
∫ +∞
β
(
1 − pFA(x))Nc−1 fχ2(x;λ,Knc) dx (3.10)
PFA|H1 = 1 − PMD − PD, (3.11)
where Nc is the number of cells in the window, fχ2(x;λ,Knc) is the PDF of the non-
central χ2 distribution of non-centrality parameter λ = 2KncTcohC/N0 and Knc degrees
of freedom. The integral in Equation 3.11 is evaluated numerically, using Simpson’s
method. The integration domain and resolution are set using a heuristic rule based on
the mode and standard deviation of the distribution. I experienced some numerical
precision issues when calculating the PDF of the non-central χ2 distribution with
large degrees-of-freedom (typically 2Knc  50), using the Scipy scientific library.
The problems were solved by using a Gaussian approximation of the non-central
χ2 distribution.
Single and multiple H1 assumptions
The flow-graph model can be completed to take into account the cases where the signal
can be detected for more than one combination of parameters. For example, this is
71
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
the case if the search space has a fine resolution and the signal is strong enough to
cross the threshold in many cells in the proximity of the best estimate. O’Driscoll (2007)
modified the flow graph to include more than one incoming transitions to the detection
state, although this method would work only for a detection in more than one window,
and not at the cell level.
To resolve the case of the detection of a signal in multiple cells, Pany (2010) proposes
to change the number of detector cells to reflect the possibility of than more than one
cell contains the correct signal. Another approach consists in calculating the average
detection statistics over several cells.
In the rest of this analysis, we assume that there is only one correct estimate of the
synchronisation parameters in the entire search space.
3.3 Acquisition model of an assisted receiver
Introduction
In this section, I detail the calculation of the probability distribution of the time-to-first-
fix. I present mostly new results, extending the results found in the literature reviewed
in the previous section. I use the same notation T(n)FA, T
(n)
D and T
(n)
hit both for the full-space
analysis and for the reduced-space analysis. Hence the distributions of these random
variables are calculated using the detection parameters (number of cells, windows,
threshold, and signal strength) corresponding to the acquisition stage in which they
appear.
Distributions in the full-space stage
The probability generating function of the time-to-first-hit of channel n, T(n)hit , is calculated
from the Equations 3.3 set to the detection parameters of channel n. We then apply the
inversion method presented above to calculate the cumulative distribution P(T(n)hit ≤ t).
At the first hit occurring within any of the Nch channels, the receiver enters the second
stage of the acquisition process, where the search space is reduced. Mathematically, the
transition time T(full)hit is:
T(full)hit = min
{
T(1)hit, . . . ,T
(Nch)
hit
}
,
which is represented by its cumulative distribution function P
(
T(full)hit ≤ t
)
. The relation
with the cumulative distribution functions of the individual acquisition times is
P
(
T(full)hit ≤ t
)
= 1 −
∏
n=1,...,Nch
[
1 − P
(
T(n)hit ≤ t
)]
. (3.12)
72
3.3. Acquisition model of an assisted receiver
As noted previously, Equation 3.12 is calculated directly from the PGF of the comple-
mentary cumulative distribution function, as shown in Equation 3.6.
Distributions in the reduced-space stage
We use a similar method for the second stage of the acquisition process. First we
calculate the generating functions of the each individual channel n using Equation 3.3,
this time with the detection parameters corresponding to the reduced-space stage. We
then invert them to calculate the distribution functions of the time to first detection,
P(T(n)D ≤ t), and of the time to the first false-alarm, P(T(n)FA ≤ t).
There are two outcomes of interest in this stage. Either the synchronisation
parameters have been correctly estimated in at least KFIX channels, which guarantee a
fix, or at least one detected signal is a false alarm, which will prevent the position fix.
We represent the random time to reach these outcomes by T(red)D and T
(red)
FA respectively.
The time to at least KFIX hits, irrespective of false-alarms and detections, is labelled
T(red)hit and offers some simplification of the expressions.
The distributions of these random variables can be calculated using a combinatorial
approach. We denote Sk(Ω), the set of k-combinations of the elements of Ω = {1, . . . ,Nch}.
For example, S2 ({1, 2, 3}) = {{1, 2}, {2, 3}, {1, 3}}.
The distribution function of T(red)hit is then
P
(
T(red)hit ≤ t
)
=
Nch∑
k=KFIX
[ ∑
σ∈Sk(Ω)
ω∈Ω\σ
P
(
T(σ1)hit ≤ t
)
× . . . × P
(
T
(σKFIX )
hit ≤ t
)
×P
(
T(ω1)hit > t
)
× . . . × P
(
T
(ωk−KFIX )
hit > t
)]
(3.13)
and the distribution function of T(red)D is
P
(
T(red)D ≤ t
)
=
Nch∑
k=KFIX
[ ∑
σ∈Sk(Ω)
ω∈Ω\σ
P
(
T(σ1)D ≤ t
)
× . . . × P
(
T
(σKFIX )
D ≤ t
)
×P
(
T(ω1)hit > t
)
× . . . × P
(
T
(ωk−KFIX )
hit > t
)]
. (3.14)
P
(
T(red)FA ≤ t
)
can be calculated with a similar combinatorial argument. However, it is
simpler to use the relation
P
(
T(red)FA ≤ t
)
= P
(
T(red)hit ≤ t
)
− P
(
T(red)D ≤ t
)
. (3.15)
73
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
We use the convention that the summation over an empty set equals 1. These equations
are very similar to the calculation of the order statistics of a collection of random
variables.
Calculation of the overall distribution
Distributions after the reduced-space stage
The acquisition over the full-space stage stops when KFIX signals out of N have been
detected, giving outcomes in two different absorbing states:
• the false alarm state FA, where at least one of the detected signals is a false alarm,
and reached with a random time of TFA dwell times; and
• the true detection state D, where all detected signals are correct detections, and
reached after TD dwell times.
The expressions of these random times are
TD = T
(full)
hit + T
(red)
D
and
TFA = T
(full)
hit + T
(red)
FA .
We use the additivity property of the probability generating function, seen in
Equation 3.1, to calculate the generating functions of TD and TFA, respectively:
UD(z) = U
(full)
hit (z) U
(red)
D (z)and (3.16)
UFA(z) = U
(full)
hit (z) U
(red)
FA (z). (3.17)
These generating functions can be easily calculated using the inversion formula of the
cumulative distribution functions presented in Section 3.1. In this case, U( f ull)hit (z) is
calculated from the distribution function of Equation 3.12, while U(red)D (z) and U
(red)
FA (z)
are calculated from the CDF of Equations 3.14 and 3.15 respectively.
Summary of calculations
Figures 3.3 and 3.4 summarise the steps that lead to the calculation of the generat-
ing functions required by Equations 3.16 and 3.17. The blocks PGF→ CDF and
CDF→ PGF represent the numerical inversion functions detailed in Section 3.1.
Inclusion of the false-alarm penalty time
We stated in the previous section that we model a false-alarm by a deterministic penalty
time of Tpenalty dwell times. Its generating function is Hpenalty(z) = zTpenalty . The signal
74
3.3. Acquisition model of an assisted receiver
Channel 1
First hit
T(full)hit =
min
{
T(1)hit, . . . ,T
(Nch)
hit
}
P(T(1)hit ≤ t)PGF→
CDF
CDF
→
PGF
U(1)hit(z)
P
(
T(full)hit ≤ t
)
Channel Nch
PGF
→
CDF
U(Nch)hit (z) P(T
(Nch)
hit ≤ t)
U(full)hit (z)
Figure 3.3: Diagram of the calculation of the theoretical time-to-first-hit at the
full-space stage.
Channel 1
Equations 3.14 and
3.15.
P(T(1)hit ≤ t)PGF→
CDF
CDF
→
PGF
U(1)hit(z)
P
(
T(red)D ≤ t
)
,
P
(
T(red)FA ≤ t
)
Channel Nch
PGF
→
CDF
U(Nch)hit (z) P(T
(Nch)
hit ≤ t)
U(red)FA (z),
U(red)D (z)
Figure 3.4: Diagram of the calculation of the theoretical time to KFIX detection and
time to one false alarm, at the reduced-space stage.
detected in the full-space stage is used to calculate the reduced spaces. We choose to
start again the acquisitions over all K channels upon entering the reduced-space stage
and the signal initially detected at the full-space stage is not tracked. This simplifies
the modelling of false alarms, as false alarms can only be detected after all the satellites
have been acquired.
The flow graph shown in Figure 3.5 represents an assisted receiver that can detect
and recover from the false alarm state, with a fixed penalty time of TPenalty dwell times.
The only absorbing state of this graph is now FIX, which represents the correct detection
of at least KFIX signals. It is reached at the random time TFIX of generating function
UFIX(z). Using the loop-reduction technique shown by Polydoros et al. (1984a), the
75
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
UFA(z)z
TPenalty
UD(z) FIX
Figure 3.5: Flow graph of the assisted acquisition process with false-alarm detection
and recovery.
probability generating function of the TTFF is
UFIX(z) =
UD(z)
1 −UFA(z) zTPenalty
. (3.18)
Finally, the PGF inversion method from Section 3.1 is applied to calculate the
probability distribution of the time-to-first-fix, P(TFIX ≤ t).
Model validation
We validated the theoretical model using Monte-Carlo simulations. The false-alarm
rate at the cell level was chosen to be 10−6, a value probably too large for a real receiver
but that generated enough false-alarm events to verify the theoretical model while
maintaining a reasonable simulation time.
The simulation results are presented in Figure 3.6 and show a good agreement with
the theoretical model.
0 200 400 600 800 1000 1200 1400 1600
Time (number of dwells)
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
P
ro
ba
bi
lit
y
of
fir
st
fix
P
fix
(t
)
Sampled
Theoretical
Figure 3.6: Monte-Carlo simulation of the TTFF. Plain line: theoretical TTFF.
Crosses: sampled TTFF from Monte-Carlo simulations with 105 runs.
76
3.4. Illustration for typical signals
3.4 Illustration for typical signals
In this section, we apply the model derived previously to a simple, simulated, acqui-
sition scenario. This provides some insight into the characteristics of the probability
distributions.
Examples of TTFF distributions
I analysed the theoretical TTFF performance by considering four simple power com-
binations. The baseline configuration was an assisted-GPS receiver in a weak-signal
environment, where eight satellites were received with a signal strength of 22 dB−Hz.
The remaining configurations considered the cases of 1, 2 or 3 stronger line-of-sight
signals received with a strength of 32 dB−Hz. The receiver comprised eight parallel
channels, each channel having twelve windows at the full-space stage and one window
at the reduced-space stage. The detector had a coherent integration period of 1 ms,
which was then squared and repeated to form 300 non-coherent integrations. The
dwell time was thus 300 ms. The detection threshold was set for a target probability of
false-alarm of 10−6. A minimum of five satellites was needed to calculate a position. A
false alarm restarted the acquisition process and generated a penalty time of 600 ms.
The TTFF distributions for these four configurations are presented in Figure 3.7.
The intermediate distributions are represented in Figure 3.8.
0 10 20 30 40 50 60
Time-to-first-fix (s)
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Pr
ob
ab
ili
ty
of
fix
0 1 2 3
Figure 3.7: Distribution of time-to-first-fix. The value in the legend is the number
of signals with a C/N0 of 32 dB−Hz, out of 4 signals. The remaining signals have a
C/N0 of 22 dB−Hz, for a total of 8 satellites.
77
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
0 2 4 6 8 10
Time since start (s)
0.0
0.2
0.4
0.6
0.8
1.0
Pr
ob
ab
ili
ty
of
hi
t
32 dB-Hz
22 dB-Hz
(a) Probability distribution of a hit on a single chan-
nel at the full-space stage.
0 2 4 6 8 10
Time since start (s)
0.0
0.2
0.4
0.6
0.8
1.0
Pr
ob
ab
ili
ty
of
hi
t
0
1
2
3
(b) Time-to-first-hit distribution at the full-space
stage for each configuration.
Figure 3.8: Intermediate distributions at the full-space stage.
Analysis
The profile of the distribution function shown in Figure 3.7 can be understood by
looking at the shape of the distribution at the full-space stage. In particular, Figure 3.8(a)
shows that the distribution function of a single satellite is composed of linear segments.
The slope of the distribution is proportional to the probability of detection, steeper
slopes corresponding to stronger signals. It can be shown that the acquisition time of
a single signal with a hybrid search strategy is in fact piece-wise linear, a result that
was established by Holmes et al. (2006), using a different method. The time-to-first-hit
distribution, calculated from Equation 3.12, is thus the product of piece-wise linear
functions, which results in a piece-wise polynomial. This polynomial aspect is shown in
Figure 3.8(b). The order of the polynomial is related to the number of signals received,
with a higher order polynomial corresponding to a larger number of satellites. The
coefficients of the polynomials are related to the detection probability of the signals.
The performance of the first acquisition stage is dependent on the strongest signals,
and not on the total number of signals. As shown in Figure 3.8(b), the performance gain
between one and three satellites with relatively stronger signals (32 dB−Hz) is small.
This is an important result for assisted-GNSS in urban environments, where only few
satellites are in line-of-sight. The number of signals is not critical for the performance
of the first stage. In this situation, it seems that minimizing the false-alarm rate at
the first stage is more important than optimizing the time-to-first-hit. Intuitively, the
multiplication of navigation signals and the parallelisation capabilities of the receivers
can guarantee that the time-to-first-hit at the full-space stage is already close to being
optimal. However, false alarms may have a severe impact on the overall performance,
so the optimisation goal at the full-stage space should be to minimize the false-alarm
78
3.5. Application to real signals
rate.
3.5 Application to real signals
The theoretical distribution of the TTFF should be compared and validated with results
from a real receiver. However, an experiment supporting this comparison is very
difficult to set up. We would need require an in-depth knowledge of the architecture
of a receiver and run several acquisitions with a GNSS constellation simulator, which
would be prohibitively time-consuming. Instead, we regulate the theoretical model
with real signal-strength measurements, taken in an office.
First, we introduce the experiment performed to measure the signal strength of
indoor signals. We then follow the design process of an assisted receiver: we optimize
the detection parameters and calculate the search space sizes, based on the assistance
data available. Finally, we present the theoretical distribution of the TTFF, as calculated
using the model from the previous sections.
Indoor signals characterisation
Data collection
The direct measurment of the C/N0 of weak signals is often not possible, since the true
synchronisation parameters θtrue can not be estimated reliably. An alternative is to
use a dual-input receiver with a common master oscillator, such as the Nordnav R30
presented by Mitelman et al. (2006). This configuration allows one antenna to be located
in an area with a strong signal, suitable for reliable estimation of the synchronisation
parameters. The other antenna can be set in a more challenging environment, where a
weaker signal is expected.
Since a common oscillator provides the frequency reference for the capture of both
signals, their relative phase is maintained. If the two antennas are close enough such
that the code phase difference at the two locations is negligible, the true synchronisation
parameters of both signals are approximately the same. A long coherent integration
time can then be used to estimate the C/N0 of the weaker signal reliably.
C/N0 estimation
A dual-input Nordnav R30 receiver was used to estimate the C/N0. One antenna was
positioned outdoors, on top of a building with a clear view of the sky. The other antenna
was located inside an office, with an obstructed view of the sky. Figure 3.5 shows the
79
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
Figure 3.9: Overview of the office used to collect the measurements.
office and the positions of the antennas. A building faced the window and there was
no direct view of the sky from the indoor antenna position.
The signal from the outdoor antenna was used to estimate the synchronisation
parameters, allowing a running 200 ms coherent integration of the indoor signal. The
C/N0 was then calculated using the moments method proposed by Pauluzzi et al. (2000)
for every satellite p = 1, . . . ,P acquired by the outdoor antenna. The C/N0 estimate was
then filtered with a moving-average filter of length 1 second.
The attenuation caused by the radiation pattern of the antenna is not taken into
account because there is generally no line-of-sight signal indoors. We made the
assumption that the indoor antenna had an isotropic pattern, even though the type of
antenna used, a patch antenna, exhibited attenuation at low elevation angles.
C/N0 results
The C/N0 measurements were gathered in six sets of about 6.5 minutes each approxi-
mately 24 hours apart. The starting time of the acquisition was changed every day so
that the satellite geometry remained the same every day. The time difference between
two GPS orbital periods and a solar day was about 3 minutes and 56 seconds. The
starting time was set at about 02:00 hours, when the building was empty of people.
The duration of the data collection was limited by the storage space available and
the processing times.
80
3.5. Application to real signals
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
C
/N
0
(d
B-
H
z)
(a) PRN 9
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
C
/N
0
(d
B-
H
z)
(b) PRN 15
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
C
/N
0
(d
B-
H
z)
(c) PRN 17
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
C
/N
0
(d
B-
H
z)
(d) PRN 18
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
C
/N
0
(d
B-
H
z)
(e) PRN 22
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
C
/N
0
(d
B-
H
z)
(f) PRN 26
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
C
/N
0
(d
B-
H
z)
(g) PRN 27
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
C
/N
0
(d
B-
H
z)
(h) PRN 28
Figure 3.10: C/N0 time series for individual satellites. Each line within one
plot represents the C/N0 recorded on a different day for the same satellite. The
beginning of every time series is set to the same sidereal time.
81
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
The C/N0 time series are shown in Figure 3.10. We observe that, for each satellite, a
common fading pattern occurs every day. This pattern is probably caused by multipath
fading. The indoor antenna is in a typical indoor fading environment. There is no
line-of-sight with the satellites and the incoming radio-frequency signal is scattered by
the building structure. The phase differences between the many scattered signals are
changing with time since the satellites are moving. The phase difference introduces
some fading when the scattered waves recombine at the receiver antenna.
The time series from satellites with PRN 28 and PRN 18 demonstrate that the signal
strength can change by more than 10 dB in the space of one minute.
The satellites with PRN 15 and PRN 27 have the strongest signals and also have the
highest elevation angles. There is no clear correlation between the elevation angle and
the signal strength for the remaining satellites.
We observe a small lag between the patterns on consecutive days for each satellite.
This could have been caused by a systematic error in the starting time of the acquisition,
which resulted in the multipath geometry not being the same every day. It was decided
not to correct this starting time offset since it was not fundamental to the rest of the
analysis.
It is clear from Figure 3.11 that there is no global correlation of the signal strength
of all satellites during one day.
Receiver parameters
The assistance scenarios are presented in Table 2.1 of Chapter 1. The search space limits
for the three scenarios are presented in Table 3.1, using the approximations provided
by van Diggelen (2002).
Table 3.1: Search space size, for each scenario.
Aiding Frequency Code phase
scenario uncertainty uncertainty
Low level ±4 kHz 1 ms
Medium level ±350 Hz 1 ms
High level ±350 Hz 20µs
In most situations, the probability distribution of the prior information is considered
to be uniform. The uniform distribution is the simplest model representing prior
information. In theory, better models could be designed for GNSS systems. For
example, timing errors are not strictly uniform and can be derived more accurately
from the clock model. Similarly, the user position prior distribution is not strictly
82
3.5. Application to real signals
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
40
C
/N
0
(d
B-
H
z)
(a) Day 1
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
40
C
/N
0
(d
B-
H
z)
(b) Day 2
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
40
C
/N
0
(d
B-
H
z)
(c) Day 3
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
40
C
/N
0
(d
B-
H
z)
(d) Day 4
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
40
C
/N
0
(d
B-
H
z)
(e) Day 5
0 50 100 150 200 250 300 350 400
Time (s)
10
15
20
25
30
35
40
C
/N
0
(d
B-
H
z)
(f) Day 6
PRN 09
PRN 15
PRN 17
PRN 18
PRN 22
PRN 26
PRN 27
PRN 28
(g) Legend
Figure 3.11: Daily C/N0 time series
83
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
uniform. However, in practice, the uniform assumptions provides good performance
and limits the complexity of the search algorithm.
Table 3.1 gives the extent of the continuous search space. Sweeping the continuous
search space is feasible but not practical, since the sequences are designed such that
the likelihood function of the synchronisation parameters is almost zero everywhere
except in a small region close to the right parameters. We thus want to discretise the
search space, with a resolution fine enough so that the correlation peak is not “missed”,
but coarse enough in order to minimise the size of the discrete search space. I chose a
code-phase resolution of ∆τ = 1/2TCA, which limited the maximum signal attenuation
to about 2.5 dB.
The frequency resolution is constrained by the coherent integration length, since
coherent integration assumes that the phase difference between the incoming signal and
the local oscillator is constant during the integration time. If the frequency resolution is
too coarse, the mismatch between the frequency of the incoming signal and the local
oscillator will introduce a systematic phase difference, which increases in time and
reduces the correlation result. When the phase difference reaches 180 degrees, new
samples added to the sum begin to cancel previous samples.
The relation between the frequency error, ∆ f , and the equivalent signal loss, L, is
found in many references, for example in Stirling-Gallacher et al. (1996), and is recalled
here:
L =
∣∣∣∣∣∣ 1Tcoh sin
(
pi∆ f Tcoh
)
pi∆ f Tcoh
∣∣∣∣∣∣2 .
Ward et al. (2006) recommends choosing the frequency resolution
∆Fmax =
2
3Tcoh
,
which limits the maximum loss to about 1.6 dB. For Tcoh = 1 ms, this corresponds to a
frequency resolution of ∆Fmax ≈ 667 Hz.
The PRN search space is already discretised, since there are only a finite number
of satellites. The resolution of this dimension of the search space generally assigns
one PRN to every receiver channel. However, some search strategies work at a lower
resolutions: the PRN sequences from more than one satellite are combined and both
signals are searched at the same time. This approach can improve the TTFF by reducing
the search space, as shown by Jan et al. (2009) but it can only work with strong signals
because of the cross-correlation properties of the PRN sequences.
84
3.5. Application to real signals
Table 3.2: Search space size, for each scenario.
Aiding Channels Windows Cells
Low level 8 12 2046
Medium level 8 1 2046
High level 8 1 40
Parallelisation capabilities
We assume that the code-phase space, for all satellites and for a given frequency offset,
can be searched in one dwell time. The search for a new code phase requires a new
signal acquisition and thus a new dwell time. This is typical of a multi-channel receiver
where the frequency-offset removal is performed in hardware and the code search is
performed by a software algorithm using FFTs.
Total number of cells and windows
We can now use the size of the search space and the parallelisation capabilities
established above to calculate the number of cells and windows. Table 3.2 shows the
number of cells and windows of the flow graph.
Optimisation of the detector parameters
Mean acquisition time
Three parameters of the GLRT detector need to be set by the designer: the coherent
integration time Tcoh, the number of non-coherent integrations KNC and the detection
threshold β. The detection threshold β is directly related to the probability of false
alarm at the cell level, pfa, at constant noise power and integration time; it does not
depend on the signal strength.
These three parameters have a direct influence on the overall acquisition performance
by increasing the probability of detection. A longer integration time and a lower
threshold directly increase the probability of detection. However, this increase in
sensitivity has an negative impact on the acquisition time: a longer coherent integration
time increases the dwell time and a lower threshold raises the false alarm rate and the
number of re-initialisations of the search.
The optimal choice of these parameters is thus a trade-off between receiver sensitivity
and acquisition time, for a given signal strength and receiver architecture. The
detector parameters can be optimized using a numerical optimisation routine. One
85
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
approach would be to calculate a statistic, such as the mean or the 90-percentile, of
the distribution of the TTFF calculated in Section 3.3, and optimize over the detector
parameters. However, this solution is not practical since the computation time of the
TTFF distribution would become too large for a numerical optimisation solution and
there is no closed-form solution is available.
However, the mean acquisition time of a single signal is known and has a simple
closed-form solution. For example, Holmes et al. (1977), and Polydoros et al. (1984a)
give the closed-form expression of the mean acquisition time for a serial search. Using
a symbolic calculation software, it is easy to derive a similar result in the case of a
hybrid search:
Tacq =
1
2PD
(
(N − 1)(2 − PD)PFA|H0 + 2PFA|H1
)
Tpenalty +
(2 − PD)N
2PD
+
1
2
, (3.19)
where Tacq is the mean acquisition time expressed in number of dwell times.
Coherent integration time
The coherent integration time can be excluded from the optimisation procedure. In a
matched-filter receiver, the maximum coherent integration is limited by the period of
the navigation bits, and the minimum integration time by the period of the spreading
code. As seen previously, the size of the frequency-offset search-space increases linearly
with the coherent integration time. O’Driscoll et al. (2008) have studied whether the gain
in receiver sensitivity was worth the increase in the search space size. They recommend
the choice of the shortest coherent integration time possible. Similarly, Anghileri et al.
(2010) considered only the shortest coherent integration time when analysing the design
of a receiver. The rest of the analysis will therefore assume a coherent integration time
of Tcoh = TCA=1 ms.
Number of non-coherent integrations and threshold
The penalty time in Equation 3.19 is different from the penalty time introduced in
Section 3.3, where the penalty time is encountered after all satellites have been acquired.
Here, Equation 3.19 gives the mean acquisition time of one satellite only. Since, for
an assisted receiver, a false alarm is detected only after all KFIX satellites have been
acquired, the acquisition time needs to be artificially increased to represent the wait for
the acquisition of the KFIX − 1 remaining satellites. Arbitrarily, I set the penalty time
to five seconds; the exact value is random and depends on the implementation of the
acquisition algorithm.
86
3.5. Application to real signals
The average C/N0 of the measurements, 25.8 dB −Hz, is used as the design point of
the detector. As seen previously, a target window-level false-alarm rate of about 10−3
leads to a cell-level false-alarm of about 10−6, for about 2000 cells.
Figure 3.12 shows a contour plot of the acquisition time of a single signal as a
function of the number of non-coherent integrations and the probability of false alarm,
for a penalty time of five seconds and over 2046 cells and 5 windows.
−9 −8 −7 −6 −5 −4
Probability of false alarm (log10)
200
400
600
800
1000
1200
1400
N
um
be
r
of
no
n-
co
he
re
nt
in
te
gr
at
io
ns
1.5
2.0
3.0
3.0
4.
4.0
5.0 5.0
5.0
10.020.0 0.5
1.0
1.5
2.0
3.0
4.0
5.0
10.0
20.0
50.0
Figure 3.12: Mean time to acquisition of a single signal, in seconds. The time scale
is not linear.
Three regions can be distinguished. At short integration time, the mean acquisition
time is increased because of the lower probability of detection. For large number of
non-coherent integrations, the probability of detection is large enough to ensure a
reliable detection of the signal. The dwell time increases linearly with the number
of non-coherent integrations. The mean acquisition time can become dominated by
the dwell time when the number of non-coherent integration is increased. Finally, the
search is slowed by an increase in the false-alarm rate, which drags the acquisition into
too many false-alarm recoveries.
I chose a target probability of false-alarms of 10−6 and 400 non-coherent integrations
for the rest of this analysis based on an examination of Figure 3.12. A more rigorous
approach would use a numerical optimisation algorithm to calculate the optimal
parameters. However I do not believe that a numerical optimisation routine would
bring a significant performance improvement.
87
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
0 1 2 3 4 5
TTFF (s)
50
100
150
200
250
300
350
Ti
m
e
si
nc
e
st
ar
to
fd
at
a
se
ri
es
(s
)
0.00
0.50
0.60
0.70
0.80
0.90
0.95
1.00
(a) Low assistance level
0 1 2 3 4 5
TTFF (s)
50
100
150
200
250
300
350
Ti
m
e
si
nc
e
st
ar
to
fd
at
a
se
ri
es
(s
)
0.00
0.50
0.60
0.70
0.80
0.90
0.95
1.00
(b) Medium assistance level
0 1 2 3 4 5
TTFF (s)
50
100
150
200
250
300
350
Ti
m
e
si
nc
e
st
ar
to
fd
at
a
se
ri
es
(s
)
0.00
0.50
0.60
0.70
0.80
0.90
0.95
1.00
(c) High assistance level
Figure 3.13: Contour plots of the distribution of the TTFF as a function of the
starting time of the acquisition, for the signal strengths recorded on day 1. The
vertical axis represents the starting time of the acquisition, where the origin is the
beginning of the measurements. The horizontal axis covers the possible values of
the TTFF. The probability of fix is indicated by the colour of the contour plot.
88
3.6. Conclusions and future work
Time-to-first-fix distribution
The TTFF distribution is not constant in a fading environment, since the strengths of the
signals from each satellite can vary. These variations affect the probabilities of detections
and thus the acquisition times of the individual satellites. The acquisition time is
increased during a fade, when the signal is weak. However, the TTFF is a function
of the combinations of the signal strength of the individual satellites. Figure 3.11
shows that the signal strengths of the individual satellites are not correlated. The
stationarity properties that can be observed on the individual satellites in Figure 3.10
are not present when all signals are jointly considered. The analysis of the TTFF is thus
not straightforward and we choose to calculate the TTFF distribution at every time
step. The evolution of the TTFF distribution as a function for the starting time of the
acquisition is represented by contour plots in Figure 3.13, for the measurements taken
on day 1.
Analysis
The overall TTFF performance of an assisted receiver can be significantly changed by a
single satellite. This behaviour is specific to assisted receivers, where the first stage of
the acquisition process relies on the acquisition of a single satellite. The second stage of
the acquisition process, that requires the acquisition of M − 1 satellites, is more affected
by the average signal strength of all satellites. This is well illustrated by looking at
Figure 3.13. We observe a first period in the TTFF distribution, of about 180 seconds,
where the TTFF has large variations and its median value is often more than 3 seconds,
for any level of assistance. Then, from about 180 seconds to the end of the data series,
the TTFF distribution becomes more regular. These two phases are understood by
looking at the C/N0 time series of all satellites, in Figure 3.11. From about 180 seconds
onwards, the signal from the satellite with PRN 27 becomes clearly stronger than the
other signals, thus reducing the acquisition time of the first satellite and hence the time
the acquisition process spends searching the full space. This satellite has the highest
elevation, more than 60 degrees and probably had the least phase difference between
the reception paths, hence there was little fading.
3.6 Conclusions and future work
In this chapter, we have seen the calculation of the time-to-first-fix for the theoretical
model of an assisted receiver. I applied the theoretical framework to real signal strength
89
3. Analysis of the time-to-first-fix in assisted-GNSS receivers
measurements, taken indoors, and I calculated the theoretical probability distribution
of the TTFF in this environment. The signal strength measurements highlighted the
fading nature of the indoor channel, with the signal strength variations of more than
15 dB. However, the TTFF distribution was more affected by the strongest signal than
the average or distribution of the other signal strengths. This is caused by the two-stage
acquisition process of assisted receivers: in a fading channel, the TTFF performance
depends more on the acquisition time of the first signal than of the remaining signals.
This work provides the theoretical bounds of the TTFF, at any arbitrary confidence
level, for a theoretical receiver model. The theoretical bounds could be used when
comparing different search strategies or during the verification and testing of navigation
receivers. Another application could to adapt the detection parameters as the search
time increases. For example, one could calculate the 90%-TTFF for different types of
environments, such as outdoors, urban, and indoors. Then, starting with the least
challenging environment, the search strategy can adapt its parameters if no fix has
been obtained after each of the theoretical bounds has been reached. Finally, the
receiver could decide to stop all GNSS signal processing after the time limit for the
most challenging has been reached, and thus save battery energy.
However, I believe the main interest of this chapter lies more in the mathematical
tools and the approach that were used in order to calculate the TTFF distribution, than
in the resulting TTFF distribution itself. The approach presented previously provides a
framework that can be used to model the acquisition processes of different types of
receivers. I focused on the theoretical model of an assisted receiver, but models of other
types of satellite navigation receivers and, more generally, CDMA systems, could be
developed.
Future work
The model assumes a slow-fading channel, where the acquisition time is shorter than
the duration of the fades. The constant detector probabilities implicitly assume that
the signal strength is constant between the entry point of the flow graph and the end
state FIX. From the channel measurements shown in Figure 3.11, this assumption is
generally valid for a static receiver. However, it is known that the crossing rate of
the fades increases for a moving receiver. The theoretical model based on constant
probability of detection may then overestimate the TTFF.
This limitation is difficult to resolve. If the channel model is known analytically, it
may be possible to calculate the detection statistics. For example, Simon et al. (1998)
have developed an extensive framework for the calculation of detection probability for
90
3.6. Conclusions and future work
many fading channel models. However, this approach still assumes that the detection
probabilities are time independent. Including the time correlation of the fading channel
within the flow-graph method is very challenging, since a general assumption is that
the probability of detection remains constant over all search windows. Corazza et al.
(2004) analysed such a method for the acquisition time of CDMA systems and noted
that the flow-graph method cannot be used in the general case of time correlation of the
detection statistics. They propose a direct calculation of the acquisition, which relies
heavily on combinatorial arguments.
More work should be put in the modeling of the detection of false alarms in assisted
receivers. The literature of flow-graph methods almost exclusively considers a fixed
false-alarm penalty time. This may not be representative of the behaviour of an assisted
receiver, depending on the algorithm or mechanism used to detect false alarms. If
the calculation are performed on a network server, as it is the case in the MS-assisted
scheme described in Chapter 1, the penalty time depends on random parameters, such
as the network load. Although I have not tried, it seems reasonably straightforward to
include a random penalty in the flow graph, by simply replacing the penalty generating
function of a deterministic random variable with its random counterpart, corresponding
for example to a uniform distribution.
The calculation of Equation 3.14 and 3.13 can become computationally demanding
as the number of channel increases. This problem is frequent in combinatorics and
it could be alleviated by calculating approximate solutions. We could, for example,
consider only the K satellites with the strongest signals, or we could group the satellites
into sets with similar signal strength and give them appropriate weights to reflect their
real cardinality. This method should give a good accuracy since the TTFF is determined
by the few satellites with the strongest signals.
91

CHAPTER 4
Time assistance from
unsynchronised cellular networks
Accurate time information can help reduce the time-to-first-fix of a GNSS receiver, thus
improving the user experience of mobile devices. One common method to obtain time
information is to use a cellular communication network, which is accessible by many
mobile devices. Many cellular networks, such as GSM networks, are unsynchronised
and operators often need to deploy dedicated measuring equipment to disseminate
the time information. However, we show in this chapter that the long-term stability
of a GSM network could be exploited to provide accurate time assistance to a GNSS
receiver, without the need for any external equipment on the network. We first review
the stability properties of oscillators and I summarise the synchronisation properties of
GSM networks. We present the results of experiments on the time stability of some
GSM networks in Cambridge, United Kingdom. We then investigate the performance
of a linear Kalman filter, combined with a two-state oscillator model, to predict the
time deviations of the GSM base stations. Finally, we exploit the long-term stability of
several GSM base stations to improve the accuracy of the local oscillator by building a
clock ensemble filter. We demonstrate that the performance of the filter could be of
practical interest to improve the time accuracy of mass-market devices.
93
4. Time assistance from unsynchronised cellular networks
4.1 Introduction
Motivation for reducing the search space
We have seen in the previous chapter that shortening the TTFF can be achieved by
reducing the search space and by increasing the parallelisation level of the acquisition
process. Massively-parallel hardware correlators, and FFT-based software correlators,
have helped increase the parallelisation of the search, to the point that the all the
synchronisation parameters of the GPS L1 C/A signal can currently be searched in
parallel in a single dwell time by some receivers. A reduced search space may not
necessarily bring a large improvement in the TTFF performance since the minimum
acquisition time is limited by the dwell time, which sets the granularity of the TTFF.
The dwell time, as seen in Equation 3.7, is independent of the size of the parallel search
windows, hence reducing the size of the search windows may not reduce the dwell
time. There are, however, at least three incentives to seek a reduction in the search
space.
Firstly, the energy consumption of hardware correlators grows with the size of the
search space, hence reducing the search space also improves the power efficiency of the
receiver. This can be a major operational and commercial advantage in mobile devices.
Reducing the search space below the size of a search window, which is fixed by the
parallelisation level of the receiver, also decreases the number of false alarms. Each
false alarm degrades the TTFF since the search has to start again, which was modelled
by the penalty time in the previous chapter.
Finally, GNSS receivers will have to deal with larger numbers of ranging signals in
the near future, thus increasing the total search space. The periods of the PRN codes of
the new ranging signals also tend to be longer, which increases the size of the search
space along the code-phase axis. A longer PRN code increases the length of the coherent
integration if the spreading rate is not increased. A longer coherent integration requires
a finer resolution in the frequency-offset search, as we have seen in Section 3.5, which
also expands the search space along the frequency offset axis. Galileo’s E1 open service
has, for example, a pseudo-noise period of 4092 chips transmitted at 1.023 Mchips/s,
compared with 1023 chips transmitted at the same spreading rate for GPS L1 C/A. The
search space for Galileo E1 open service is thus 16 times larger than for the GPS L1 C/A
signal. It is possible that software receivers may have difficulties coping with such
an increase in search-space size and the “real estate” cost of adding more hardware
correlators may become prohibitive. Furthermore, in all cases, reducing the search
space always minimizes the computational load and thus improves the power efficiency
94
4.1. Introduction
of the receiver. This can be of prime interest in personal mobile devices.
Prior information, such as approximate position and time, can help reduce the size of
the search space and is commonly used in assisted-GNSS receivers. This chapter focuses
on aiding using time information only. Most mobile receivers already implement some
form of frequency control, often called frequency steering, by locking the local oscillator
on the RF carrier. Frequency steering does help time aiding, but the time reference of
the local clock needs to be initialized. Frequency steering was designed to tune the
mobile station to the RF frequency of the channel and not to provide time assistance.
There is no guarantee of synchronisation between the frequencies of different base
stations. Frequency steering also requires the mobile always to listen to a specific
channel, which is resource demanding. A mobile receiver could manage the battery
energy more efficiently by turning itself to an idle state for short periods of time The
frequency steering of the local clock may be disrupted during the energy-saving periods
and time-keeping may be lost.
Previous work
Accurate time is necessary to support the operation of assisted-GNSS receivers and to
develop cellular-based positioning systems. Many cellular networks, such as the GSM
networks, are not synchronised by design and providing accurate time information
to the network users can be challenging. Cellular network operators have deployed
dedicated equipment, such as location measurement units (LMUs), to measure the
time deviations of the signals in unsynchronised networks. These units are composed
of receivers that measure the time fluctuations of the signals transmitted by the BTSs
within their ranges, and calculate the time offsets in reference to a stable clock, such as
a GPS-disciplined oscillator for example. The time information is then broadcast to the
mobile users, a process generally called time aiding or time assistance. The installation and
maintenance costs of this equipment can be prohibitive and an alternative, stand-alone,
solution is desirable. Furthermore, the coverage of a LMU is limited to a few cells and
operators may decide to deploy measuring units only over a limited portion of their
networks in order to reduce costs. Therefore, mobile devices should not expect that
time assistance is available everywhere.
Favey et al. (2001) measured the time stability of the signals from two GSM base
stations. They observed that one base station was highly stable while the time deviations
of the signals coming from the other BTS had periodic variations. Faragher (2007)
further studied the time stability of base stations in three different GSM networks. His
experimental results showed that the stability of some base stations was nearly as good
95
4. Time assistance from unsynchronised cellular networks
as that of an atomic frequency standard. He also observed some periodic variations of
the time deviations in some networks.
These experimental measurements show that the time stabilities of GSM signals
are variable across networks. However, the results suggest that some unsynchronised
networks are stable enough to provide fine-time aiding. How to provide fine-time aiding
from time signals in an unsynchronised network was investigated by Duffett-Smith
et al. (2007) and described in the patent “Transfer of calibrated time information in a
mobile terminal” (2010). They simulated a GSM network composed of clocks with noisy
time biases and frequency drifts, which were representative of the stabilities observed
during experiments. The results of their simulations showed that the local clock of a
mobile receiver could keep track of the time with an accuracy better than 200 ns over a
period of 300 min.
I extended the experimental work undertaken by Faragher (2007) by adding a
GPS receiver to improve the long-term accuracy of the measurements. I focused on
the estimation of the network clock parameters and I investigated how the stability
of the networks could benefit a mobile GNSS receiver. All the results are based on
measurements on real networks.
4.2 Overview of GSM signals
The signals used by a mobile device to communicate with a GSM network are described
in the GSM specifications. These standardisation documents can be complex and
I summarise here the parts relevant to the experiments. I also present information
collected about the core networks of GSM service providers, which are not accessible
for experimentation but may have an influence on the time stability.
Air interface specifications
The air interface describes the communication layers between a mobile station (MS),
typically a mobile phone, and a base transceiver station. The physical layer of the air
interface is specified in 3GPP (2004, TS 05.01). It is composed of physical channels on
which symbols modulated by Gaussian minimum shift keying (GMSK) are transmitted
at a rate of 1625/6 ≈ 276.833 kbit/s. The frequency band of a channel is specified by its
absolute radio-frequency channel number (ARFCN), whose conversion to frequency
is given in 3GPP (2005b, TS 05.10). The channels are allocated a 200 kHz-bandwidth
and the effective bandwidth of the GSM signal is about 135 kHz after modulation and
impulse filtering.
96
4.2. Overview of GSM signals
The GSM standard specifies several logical channels that are required for the operation
of the network. The logical channels can be broadly divided into two categories,
according to their usage: control channels and traffic channels. Control channels are
used to support the connection of the user to the network, whereas the traffic channels
carry the useful information, such as voice or data. Some channels are broadcast to all
mobile stations whereas other channels, such as the traffic channels, are dedicated to
transmissions with a specific user only.
The standard 3GPP (2004, p. 05.01) specifies that access to the physical layer is a
combination of time-division multiple access (TDMA) and frequency-division multiple
access (FDMA). These multiple access techniques are required to allow multiple
communications over the same physical channel and to provide the structure for the
logical channels. The TDMA part is a hierarchy of data units of various specific lengths.
The most elementary unit is the slot, of duration 15/26 ms ≈ 576.9µs. Eight slots form
a frame, of duration 60/13 ms ≈ 4.62 ms, and 51 frames form a control multi-frame, of
duration 3060/13 ms ≈ 235.4 ms. A super-frame consists of 26 multi-frames and a
hyper-frame is composed of 51 super-frames. The hyper-frame is the longest element in
the GSM standard and has a period of about 3 h 28 min 54 s. The spectrum usage is
optimised by FDMA, which can balance the communications over different frequency
channels.
Under TDMA and FDMA, the logical channels are assigned specific time slots,
frames, and frequencies by the GSM standard. The access to the channels and the
allocation of the mobile stations to specific channels are both controlled by the BTSs
and can be changed to adapt the radio link to the available resources. A mobile station
is connected to only one BTS, which is called the serving base station. This does not
prevent a mobile station from listening to other BTSs, but the GSM standard mandates
that the access to the network is performed by only one BTS. Most control channels
have fixed channel and frequency allocations, such that the mobile station “knows”
how to discover new base stations, for example when the device is turned on or when
the mobile station moves and becomes in range of a BTS offering better signal than the
serving BTS.
Synchronisation of GSM networks
The multiple-access techniques TDMA and FDMA rely on the synchronisation of the
mobile receiver and its serving BTS. However, the overall operation of a GSM network
is unsynchronised and the time and frequency references are not synchronised between
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4. Time assistance from unsynchronised cellular networks
BTSs. We review the synchronisation signals carried over the air interface and the
possible synchronisation mechanisms of the core network.
The synchronisation channel (SCH) is used to achieve time synchronisation between
the serving BTS and the mobile station. It transmits synchronisation bursts (SB) that
contain the extended training sequence (ETS) and an encoded frame number (FN). The
ETS is a known pattern of 64 bits, specified in 3GPP (2003a, TS 05.02) that has good
correlation properties. The frame number contained in the synchronisation burst is
encoded by a convolutional code whose parameters are given in 3GPP (2005a, TS 05.10).
The synchronisation bursts also contain the base station identity code (BSIC), which
is a 6-bit number identifying a base station. The SCH is a downlink control channel
that is physically broadcast on the beacon frequency channel of every base station. The
acquisition of a new BTS by a mobile station, for example when it enters the range of a
new cell, is very similar to the acquisition of direct-sequence spread-spectrum signals
covered in Chapter 2. The mobile station acquires the new BTS by correlating the signal
on the beacon frequency channel with a replica of the ETS. Once the correlation result
crosses a threshold, the mobile station can decode the information contained in the
synchronisation burst. This gives the frame number that is required to synchronise
the MS within the stream of frames of the channel. The position of the ETS correlation
peak gives finer time synchronisation, which is required to align the transmission with
the correct TDMA time slots.
The frequency of the RF carrier is controlled in order to support FDMA and to
avoid interferences between adjacent radio channels. The frequency correction channel
(FCCH) is used to lock the frequency of the MS to the frequency of the BTS. The
FCCH consists of frequency correction bursts (FB), during which only the RF carrier is
transmitted. (In reality, since differential pre-coding is applied before transmission, the
data content of the burst is an alternating 0-1 pattern.) The frequency stability at the
BTS shall be better than 0.05 ppm according to 3GPP (2003b, TS 05.10), for both the RF
carrier and the timebase used to generate the synchronisation bursts.
GSM networks are not synchronised networks; they require synchronisation only
between a mobile user and its serving base station, in order to implement TDMA.
The internal counters, for instance the frame number, and the transmission time of
the ETS, are not synchronised with any network time reference, but only with the
local BTS oscillator. There is a provision in 3GPP (2003b, TS 05.10) to implement a
pseudo-synchronised network to improve the call handovers between base stations;
however the extent of the deployment of this feature is unknown.
The air interface is only one element in the networked architecture of GSM networks.
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4.2. Overview of GSM signals
Many BTSs are typically connected together to a common base station controller (BSC),
which is in turn connected to the core network of the operator. The core network,
also called the network switching subsystem (NSS) in the GSM standard, provides
the essential tasks of call management, signalling, and access to the public switched
telephone network (PSTN). The stability of the core network and the stability of the local
network around a BSC may affect the stability of the air interface. Some of these network
elements may be synchronised, in particular if they are based on the synchronous digital
hierarchy (SDH) protocols, which require precise time alignments of the transceivers.
Some commercial brochures mention the compatibility of GSM equipment with the ITU
standard G.823, whose long-term stability requirement is a maximum time deviation
of 18µs for a typical E1 link of an SDH network at 2.048 Mbit/s.
It is often claimed that improving the stability of the frequency reference at the
BTS improves the overall GSM network performance. Bregni et al. (2003) performed
experimental measurements of the call handover between two BTSs whose relative
frequencies were varied between 0.1 ppm and 10 ppm. The number of calls dropped
during the base stations handovers was not affected, although the study was too small
to be conclusive. However, they showed a significant degradation of the speech quality,
measured by the amount of speech clipping and the standardised “mean opinion
score”.
Overview of W-CDMA and LTE networks
Although not central to the argument of this chapter, we review the expected perfor-
mance of third and fourth generation cellular networks.
The synchronisation and timing requirements of third and fourth generation
networks are not fundamentally different from GSM networks, although they are based
on different technologies. The 3GPP is promoting two cellular network technologies:
the Universal Mobile Telecommunication System, which is third generation network
that is already widely deployed; and the Long-Term Evolution, a fourth generation
network. The air interface of UMTS is called UMTS Terrestrial Radio Access (UTRA),
and can be of two types: UTRA-time division duplex (UTRA-TDD), for which two-
way communication is achieved by time-sharing a common physical channel, and
UTRA-frequency division duplex (UTRA-FDD, more commonly named W-CDMA
for wideband CDMA), where two distinct frequencies are used. The 3GPP specifies
frequency accuracy requirements, for both standards, that are similar to the requirements
for GSM networks. The 3GPP also specifies an global time stability of at least 2.5µs
for the TDD implementation. I have not been able to identify any major UTRA-TDD
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4. Time assistance from unsynchronised cellular networks
networks in Europe and it seems that W-CDMA has been more widely deployed globally.
LTE has a provision for broadcasting multimedia, which requires time synchronisation
over designated areas; however, the implementation is optional.
However, many factors could improve the performance of time assistance in
third and fourth generation networks. Wider signal bandwidths and improved
equipment to cope with increasing traffic may improve the accuracy and the stability
of time information. The channel bandwidth of W-CDMA is about 5 MHz and the
bandwidth of LTE channels is flexible from 1.4 MHz to 20 MHz (with a transmitted
bandwidth of 1.08 MHz to 18 MHz). A wider bandwidth improves the precision of time
measurements, as seen in Section 2.1, although it does not mean that the local oscillator
is more stable. On the network side, the trend in the industry seems to be to move
away from circuit-switched networks and towards packet-routed networks. As the
traffic increases, operators need to invest in their core networks and the current circuit-
switched architectures seem too costly and not flexible enough. In a circuit-switched
network, a path, or circuit, is established between the two end points for the duration of
the communication. In packet-switched networks, the communication is performed by
packets of information and each packet carries the address of the end point; the exact
paths used by the packets are not specified and can freely change. Packet-switched
networks have typically more flexible synchronisation requirements, although this is not
guaranteed since packet transmission, such as the Internet Protocol (IP), can be carried
on SDH synchronised networks. The problem of synchronisation over packet-based
unsynchronised networks, such as IP over Ethernet networks, is under investigation
by equipment manufacturers. Two technologies have been developed to this end: the
synchronous Ethernet (SyncE) specifications, which sets synchronisation and stability
requirements on the Ethernet protocol; and the precision time protocol (PTP) standard
IEEE 1588v2, which allows sub-microsecond synchronisation over unsynchronised
networks, such as Ethernet.
The impact of these factors on the stability and the synchronisation of modern
cellular networks is still unclear. On the one hand, Hann (2012) shows in an industry-
sponsored white paper that better synchronisation will be necessary. On the other hand,
there is no requirement for global timing stability or network-wide synchronisation in
the specifications of UMTS and LTE, except for the multimedia broadcasting option
of LTE. The real synchronisation performance of these networks will have to be
characterised experimentally. I did not investigate W-CDMA and LTE networks any
further in this work.
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4.3. Oscillator models and time estimation
4.3 Oscillator models and time estimation
Stability and instability of oscillators
Oscillators are electronic devices that produce a signal at a known, accurate, frequency.
They are composed of a feedback loop built around a resonating element, whose natural
oscillating period is precisely known. We will encounter two types of resonating
elements in this thesis. Atomic frequency reference standard exploits the electronic
transitions between known energy levels of atoms as their resonating elements. Ru-
bidium frequency reference standards are a popular choice for measuring equipment,
although their costs and size constraints make them less common in mass-market
devices. Piezoelectric crystals, such as Quartz crystals, are often used as the resonating
element in cheaper and more compact oscillators. Crystal oscillators are sensitive
to temperature changes and different mechanisms are used to deal with changing
temperatures. Two popular types of crystal oscillators with temperature corrections
are the temperature-compensated crystal oscillators (TCXO) and the oven-controlled
crystal oscillators (OCXO). TCXOs measure the component temperature and adjust
the frequency in the feedback loop as a consequence. OCXOs measure and adjust the
temperature of the component, using a heating element.
The most important quality of an oscillator is, arguably, its stability over time. But
even the best atomic standards are not perfectly stable. Statistical methods to evaluate
the stability of oscillators have been designed since the development of atomic clocks,
in the 1960s. It is important to remember that the Coordinated Universal Time (UTC)
cannot be observed directly, hence the performance of an oscillator is always measured
in comparison with another oscillator. The measurements of instabilities are generally
the phase differences, φ(t), between two oscillators under study, both of nominal
frequency f0, measured at time t in the reference oscillator time scale. One can also
calculate the time deviation x(t) = φ(t)2pi f0 from the phase difference and the normalized
frequency deviation y(t) = d x(t)d t .
The power spectral density (PSD) of the normalized frequency deviations can be
used to characterise the statistics of the instabilities in the frequency domain. The
different sections of the PSD plot show the noise processes involved in the instabilities.
These are:
• a white Gaussian phase noise, of PSD slope f 2;
• a flicker phase noise, of PSD slope f ;
• a white Gaussian frequency noise, which is also a random walk phase noise, of
PSD slope 1;
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4. Time assistance from unsynchronised cellular networks
• a flicker frequency noise, of PSD slope 1/ f ;
• and a random walk frequency noise, of PSD slope 1/ f 2.
The flicker noise process, or pink noise, is a type of noise process whose power spectral
density follows a 1/ f function. Its origin is generally less understood than Gaussian
noise or random walk noise, although it has been observed in many types of oscillators.
Flicker noise can cause some statistical estimators to diverge, for example when
estimating the autocorrelation function. The lack of an estimator of the autocorrelation
function can be a problem since it is generally more convenient to characterise the
stability of an oscillator over a period of time than in the frequency domain. The
Allan variance, or the Allan deviation, was introduced by Allan (1966) to provide a
measure similar to the autocorrelation function, which can be used to study the noise
characteristics of clocks in the time domain. It is defined as the two-sample variance of
the normalized frequency,
σ2y(τ) =
1
2
〈(
y(t + τ) − y(t)
)2〉
,
and can also be expressed using the time deviations as
σ2y(τ) =
1
2τ2
〈[(
x(t + 2τ) − x(t + τ)
)
−
(
x(t + τ) − x(τ)
)]2〉
,
where 〈.〉 is the expectation operator.
This Allan variance can be estimated from N samples of the time deviations
measurements by the sample Allan variance, defined as
σˆ2y(τ) =
1
2(N − 1)τ2
N−1∑
k=1
(
x(kTs + 2τ) − 2x(kTs + τ) + x(τ)
)2
,
where Ts is the sampling period of the input signal.
Table 4.1: Characteristics of the noise processes in the power spectral density and
the Allan variance plots, using log-log axes.
Type of noise Allan variance PSD
White phase noise ∝ τ−2 ∝ f 2
Flicker phase noise ∝ τ−2 ∝ f 1
White frequency noise ∝ τ−1 ∝ 1
Flicker frequency noise ∝ 1 ∝ f−1
Random walk frequency noise ∝ τ ∝ f−2
The slopes of the noise processes are presented in Table 4.1, both for the Allan
variance and the PSD. We can see in this table that the standard Allan variance cannot
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4.3. Oscillator models and time estimation
differentiate between the flicker and white phase noise. The modified Allan variance has
been proposed to take into account the flicker noise. As seen later, the model considered
in this chapter simply ignores the flicker noise, thus the modified Allan variance is not
investigated any further.
The stability is often quantified by the h-parameters of the oscillator, which are
related to the magnitude of the power spectral densities of the different noise processes.
The standard IEEE STD 1139 specifies the relationship between the total power spectral
density, Sy( f ), and the h-parameters:
Sy( f ) =
2∑
α=−2
hα f α.
Table 4.2 is reproduced from Brown et al. (1997, chapter 11) and shows the typical
noise parameters of three types of oscillators. The long-term stabilities of the oscillators
are determined by the white frequency noise, characterised by the h−2 parameter.
Table 4.2 shows the magnitude of the noise power spectral density changes by about
nine orders of magnitude between a Rubidium frequency standard and a TCXO.
Rubidium oscillators are typically an OCXO locked onto the frequency of a Rubidium
frequency standard. The short-term stability of Rubidium oscillators is thus the stability
of the OCXO oscillator.
Table 4.2: Noise parameters of typical oscillators.
Oscillator h0 h−1 h−2
TCXO 2 × 10−19 7 × 10−21 2 × 10−20
OCXO 8 × 10−20 2 × 10−21 4 × 10−23
Rubidium 2 × 10−20 2 × 10−24 4 × 10−29
The oscillator sensitivity to the environmental conditions is perhaps as important
as its stability in ideal conditions. Even the temperature compensation of TCXO
and the temperature control of OCXO may not be enough to guarantee the same
stability performance across a wide range of operating temperatures. The sensitivity to
temperature of a typical mass-market TCXO is between 0.1 ppm/◦C and 0.5 ppm/◦C.
Oscillator models
We now wish to build an oscillator model that includes the noise processes discussed
previously. We adopt a state-space approach, which is compatible with many forms of
recursive Bayesian estimation, such as the Kalman filter. Van Dierendonck et al. (1984)
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4. Time assistance from unsynchronised cellular networks
treated this approach in detail and proposed the two-state oscillator model, which is
frequently used in the GNSS literature.
The flicker noise cannot be represented in a state-space model with additive
Gaussian noise. We can understand this problem by remembering that the state-space
representation of a linear system is, informally, an expression of the linear relationships
between the state vector and its first derivatives, with additive Gaussian noise (see
Section 2.1). It can also be viewed as the matrix representation of a set of first-order
linear stochastic differential equations with Gaussian noise. The Gaussian noise term is
applied to the first-order derivative of the state vector and is then integrated when the
system is solved for the state vector. The integration of the Gaussian noise leads to a
random-walk noise, thus informally “skipping” the flicker noise model. This results is
well-known and several authors have proposed different approach to treat the flicker
noise in state-space models. Van Dierendonck et al. (1984) approximate the flicker noise
contribution by increasing the variance of the other noise and Davis et al. (2005) model
the flicker noise by a linear combinations of Markov processes.
Several models of oscillators have been proposed and we consider a simple model
with a state vector of two elements. The time deviations are modelled as a linear
combination of an integrated random walk and a random walk. We use a state-space
representation with the state vector x(t) =
[
b(t),
.
b(t)
]ᵀ
, where b(t) is the time deviation
and
.
b(t) the clock drift rate. The evolutions of the oscillator are described by the linear
equation:
x(tk+1) =
1 tk+1 − tk0 1
 x(tk) + w(tk+1 − tk), (4.1)
where w(t) is a Gaussian random noise process with covariance matrix Q(∆t) =〈
w(∆t)w(∆t)T
〉
.
The process covariance matrix Q (∆t) contains information about the noise charac-
teristics of the oscillator. We have already seen that the h-parameters characterise the
instabilities in the frequency domain. Similarly, the diffusion parameters q characterise
the instabilities in the time domain. The literature about the relationship between the
h-parameters and the diffusion parameters, and between the h-parameters and the
process covariance matrix, is at times confusing. Van Dierendonck et al. (1984) derived
the process covariance matrix from the Allan variance parameters and their result was
later corrected by Brown et al. (1997). Zucca et al. (2005) calculated that the corrected
covariance matrix, Q, is
Q (∆t) =
q0 ∆t + q−2 ∆t3/3 q−2 ∆t2/2q−2 ∆t2/2 q−2 ∆t
 , (4.2)
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4.4. Time stability of the GSM air interface
where ∆t is the time interval of the process evolution and q0 is the coefficient of the
random walk of the phase noise process and q−2 the coefficient of the random walk of
the frequency noise process.
We can identify the q-parameters from the previous expression with the frequency-
domain h-parameters. Pany (2010) states that the process covariance matrix given by
Van Dierendonck et al. (1984) in terms of h-parameter is incorrect. He recommends the
expression given by Brown et al. (1997),
q0 = h0/2 and (4.3)
q−2 = 2pi2 h−2. (4.4)
Chaffee (1987) presented similar results for a three-state oscillator model, which
includes a random-walk frequency drift, or frequency ageing. The noise density of the
random-walk frequency noise was further corrected by Zucca et al. (2005), who also
show that the Allan variance of an oscillator with a random-walk frequency drift is not
stationary and depends on a reference time.
4.4 Time stability of the GSM air interface
In this section, we measure the times of arrival of the synchronisation signals from
several GSM base stations in Cambridge, United Kingdom. Faragher (2007) made
similar experiments to measure the stability of the BTS clocks and I repeated his
experiments to obtain recent measurements. I extended his experimental arrangement
by adding a GPS software receiver, which provided UTC time measurements.
Experimental apparatus
The experimental arrangement consisted of a GSM receiver, a GPS receiver, a syn-
chronisation mechanism, and data storage units. It is schematically presented in
Figure 4.1.
An IFR 2319e signal analyser received the GSM signals. The input of the analyser
was connected to an outdoor GSM 900/1800 MHz monopole antenna via a low-noise
amplifier (LNA) with an approximate gain of 23 dB. The signal analyser had a general
purpose interface bus (GPIB, normalised as IEEE-488) which could be accessed from a
remote computer. A laptop computer was used to send GPIB configuration commands
to the signal analyser. The signal analyser stored the GSM samples in its internal
memory and a computer equipped with a National Instrument NI AT-DIO-32HS
digital input/output card regularly retrieved the samples using a dedicated high-speed
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4. Time assistance from unsynchronised cellular networks
IFR
Input 2
Input 1
Counter
TriggerExt Osc
Nordnav-RX
GPIB
NI interface
TCP/IP
PC 1
Network
PC 2
RPC
5 s
1 MHz
GPS RF
GSM RF
Rb
TCXO
storage
Figure 4.1: Schematic diagram of the experimental arrangement using a Rubidium
frequency reference standard. The diagram does not represent the real physical
location of the components.
parallel interface. The analyser’s oscillator was locked to an Efratom FRK-H Rubidium
frequency reference standard. I did not need the stability offered by the Rubidium
frequency standard since the GPS time was also measured, but having a stable frequency
between GPS measurements facilitated the signal processing.
The GPS receiver was a Nordnav R30 receiver, developed and described by Mitelman
et al. (2006), which was controlled by a laptop computer running a dedicated program.
The receiver was connected to a GPS patch antenna via a LNA with a gain of about 20 dB.
The receiver did not have an external oscillator input and I left the internal TCXO
oscillator as its frequency standard.
A trigger signal synchronised the recording of the GPS and GSM signals. It was
generated by the digital counter of a Measurement Computing PC-CARD-D24/CTR3
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4.4. Time stability of the GSM air interface
card. I configured the counter to provide a square wave of 5-second period when
connected to the Rubidium frequency standard. The trigger signal was connected to the
trigger input of the IFR 2319e. The Nordnav R30 GPS receiver did not have an external
trigger input. It had two inputs synchronised by a common clock and I bypassed the
RF frontend of the second input so that it accepted the trigger signal. A program was
run during post-processing to synchronise the GPS measurements acquired on the first
input with the trigger signal on the second input. The GPS receiver was started before
the trigger signal was sent, so that the transition of the trigger signal was always among
the samples recorded on the second input of the receiver.
The receivers could not acquire data continuously because of data storage and
bandwidths issues. I developed programs in C++ to control the sequencing of the
measurements. The laptop computer A ran a program to control the GSM analyser via
its GPIB interface. The laptop computer A also ran the software provided by Nordnav
to control the GPS receiver. I developed a dynamic-link library that was plugged in
the Nordnav software and that provided a software trigger to start and stop the GPS
receiver. The computer B ran a program that configured the counter, kept track of the
list of GSM beacon channels to measure, and called other programs using XML-RPC to
sequence the recording events. XML-RPC is a network protocol to call procedures on
remote machines and I used the XmlRpc++ implementation (see Morley, 2003).
A software timer on the computer B started the recording sequence at intervals of
about 7 minutes. This period was chosen as a compromise between the high sampling
rate desired for accurate signal analysis and the low total number of samples imposed
by the storage space and processing time limitations. The recording sequence was as
follows:
1. The computer B transmitted the new beacon channel frequency in the list to the
laptop computer A. It then waited for new samples on its digital input/output
interface.
2. The laptop computer A configured the IFR 2319e signal analyser with the new
frequency and armed the trigger. It sent an acknowledgement message to the
computer B.
3. The trigger signal started the acquisition and the samples were transferred to the
computer B. The laptop computer A also sent a verification message to ensure the
RF frequency, set by computer A, was the one corresponding to the acquisition
requested by computer B. This avoided skipping beacon channels because the
acquisition buffer of the signal analyser had overflowed.
4. The laptop computer B saved the samples contained in the digital input/output
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4. Time assistance from unsynchronised cellular networks
buffers and repeated the sequence with the next beacon channel frequency.
The laptop computer A also started the GPS receiver for a period of 30 seconds at
the beginning of the sequence. The files were automatically transferred to a network
storage drive at regular intervals.
The base sampling rate of the IFR 2319e signal analyser was 65.28 MHz and an IQ
decimation ratio of 32 was chosen to maximise the recording length. This gave an
effective IQ sampling rate of 2.040 MHz. The buffer was limited to 106 IQ samples, which
was about 490 ms of baseband samples at the chosen sampling rate. The synchronisation
burst was sent five times in a multi-frame, resulting in either ten or eleven extended
training sequences available in each sampling buffer. The Nordnav GPS receiver
recorded real 4-bit samples at the intermediate frequency (IF) of 4.1304 MHz and with
a sampling rate of 16.3676 MHz. I established a list of the beacon frequencies of twelve
different base stations, although the signal strengths of some of these BTSs were not
strong enough to be fully exploited.
Calculating the times of transmission of the GSM synchronisation bursts
I used the information contained in the synchronisation bursts to calculate the times of
transmission of the GSM signals, t(GSM)i (t), given in the time scale of the base station i
and measured at the instant t in the local time scale. The TOT was determined from the
position of the ETS in a burst and from the frame number, which identified the position
of the burst within a hyper-frame. The GSM standard does not define any time origin
of the signals; the only time origin we can extrapolate from the information provided
in the synchronisation burst is the origin of the current hyper-frame. The time origin of
the TOT measurement was thus the beginning of the hyper-frame containing the burst.
The first operation required to estimate the TOT is to measure the arrival time
of the ETS within a time slot. I used a method similar to the estimation of the code
phase of a pseudo-noise signal presented in Section 2.1. The correlation function was
calculated with the computationally efficient FFT-method, described in Section 2.3
and I up-sampled the baseband signal ten times in order to improve the resolution of
time measurements. Upsampling improves the accuracy only if the signal bandwidth
is perfectly limited to the Nyquist sampling frequency of 2.040 MHz. In practice,
the signal bandwidth was affected by multipath and interference, so the accuracy of
the time measurements on the upsampled signal may not have been as high as ten
times the original accuracy. However, I upsampled the signals as it provided a better
resolution that may have been useful when the results were filtered. I calculated the
cross-correlation function of the upsampled signal with an upsampled replica of the
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4.4. Time stability of the GSM air interface
ETS, in order to detect the signal using the likelihood-ratio test presented in Section 2.2.
I rejected the measurement if the squared-magnitude of the correlation peak was below
a threshold and then proceeded with the next burst. Otherwise, I measured the position
of the correlation peak relative to the burst, which gave the fractional part of the TOT,
or the TOT modulo one time slot.
The second operation is to calculate the position of the synchronisation burst within
a hyper-frame. I estimated the channel impulse response from the correlation profile of
the ETS sequence with the original (not up-sampled) signal. I then filtered the signal
with a channel equalizer configured for the channel impulse response. The equalization
step was not strictly needed but it provided a better error-free decoding rate for
weak signals. The encoded bits of the burst were recovered and the convolutional
encoding was removed by passing the bits through a Viterbi decoder of rate 1/2 and
with polynomials D4 + D3 + 1 and D4 + D3 + D + 1, where D is the delay operator. The
frame number was reset at the beginning of every hyper-frame, which occurred about
once every 3 h and 30 min. The time of transmission of the burst was given by the
frame number with a granularity of one time slot. (The frame number had a granularity
of a TDMA frame, but the synchronisation burst always occupied the same time slot
within a frame, hence the granularity was a time slot). The fractional TOT, calculated
from the position of the correlation peak, was added the TOT of the time slot to give
the time of arrival, t(GSM)i (t). I also decoded the BSIC contained in the synchronisation
burst, which was compared with the theoretical BSIC of the BTS in order to discard the
bursts that have not been decoded properly.
I developed a software GPS receiver to retrieve the GPS system time from the
samples recorded by the Nordnav R30 receiver. The navigation data was decoded in
order to detect the preamble and the time-of-week words. I did not decode the entire
ephemeris; it was downloaded from the International GNSS Service (IGS, see Dow et al.,
2009) instead. The positions of the TOW words gave the pseudo-range measurements
for the position equation. I further constrained the position equation by fixing the three
position variables to the position of the GPS antenna, estimated from a 12 h-position
average. I solved the position equation for the receiver time bias, which was the only
remaining unknown. The equation was over-constrained since there were typically six
to eight pseudo-range measurements and only one unknown variable.
The decoding of the synchronisation bursts was not free of errors, even though
the Viterbi algorithm corrected most of them. An error in decoding one of the bits
encoding the frame number could introduce large time-of-arrival errors. I developed a
heuristic algorithm to detect large outliers, based on the comparison between the time
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4. Time assistance from unsynchronised cellular networks
of transmission of a given burst and the median time of transmission of its neighbours.
The outliers were then removed and excluded from further processing.
I observed two time jumps of about−1.2 s and 2.7 s in all the time measurements, the
first one appearing about 16 s after the start of the experiment and the second occurred
about 30 min later. I did not observe the same jumps on the GPS measurements, which
therefore excluded a problem with the frequency reference standard. The problem did
not seem to have been caused by the counter either, since the durations of the jumps
were unrelated to the period of the trigger signal or any power-of-two multiple of the
base timing signals. The delay appeared simultaneously on all the BTS measurements,
thus making a single BTS unlikely to be the cause of the problem. The remaining
possibility was a problem of signal acquisition by the IFR 2319e. It was, however,
surprising that the same delay appeared at the same time on all frequencies and lasted
for several acquisitions. A problem with the triggering or the acquisition buffer of the
IFR 2319e should have lasted for only one acquisition. I was not able to determine
the origin of these jumps. So I corrected them by introducing artificial offsets in the
measurements. I repeated the experiment for the work presented later in this chapter,
in Section 4.6, and I did not observe any jumps in that experiment.
I estimated the ageing of the Rubidium frequency standard from the time difference
between the first and last GPS measurements, t(GPS)(t1) − t(GPS)(t0), and the same time
difference according to the Rubidium frequency standard, t1−t0. I estimated a frequency
ageing of 3.31 × 10−10, which was within the specifications of the equipment. I applied
this correction to all the measurements derived from the Rubidium frequency standard,
for example the times of arrivals of the GSM signals.
Measured time deviations of GSM synchronisation bursts
The timing results were filtered by a Kalman smoother to remove a few remaining
outliers. The Kalman smoother was not used as a statistical estimator but simply as a
tool to remove the remaining outliers and produce a better visualisation of the results. I
chose a two-state oscillator model with a large process noise variance, corresponding to
the h-parameters h0 = 10−16 and h−2 = 10−17. The large process noise variance ensured
that the filter kept the information from the measurement intact while rejecting the
largest outliers. I also considered a simple random-walk process model but the two-state
oscillator model had the advantage of also estimating the frequency drifts. I tried other
values of the h-parameters to make sure the smoother was neither over-fitting nor
smoothing the results too much.
110
4.4. Time stability of the GSM air interface
0 10 20 30 40 50 60 70 80 90
Time (h)
−4
−2
0
2
4
Ti
m
e
de
vi
at
io
n
(µ
s)
(a) BTSs 1a, 1b, 1c, and 1d
0 10 20 30 40 50 60 70 80 90
Time (h)
−200
−100
0
100
200
300
400
Ti
m
e
de
vi
at
io
n
(µ
s)
(b) BTS 2
0 10 20 30 40 50 60 70 80 90
Time (h)
−4
−2
0
2
4
6
Ti
m
e
de
vi
at
io
n
(µ
s)
(c) BTS 3
0 10 20 30 40 50 60 70 80 90
Time (h)
−5
0
5
10
15
20
25
30
35
Ti
m
e
de
vi
at
io
n
(µ
s)
(d) BTS 4
0 10 20 30 40 50 60 70 80 90
Time (h)
−10
−5
0
5
10
15
Ti
m
e
de
vi
at
io
n
(µ
s)
(e) BTS 5
0 10 20 30 40 50 60 70 80 90
Time (h)
−25
−20
−15
−10
−5
0
5
Ti
m
e
de
vi
at
io
n
(µ
s)
(f) BTS 6
Figure 4.2: Estimated times of arrival of the synchronisation bursts from the BTSs.
The results have been smoothed, centred by removing the arrival time of the first
sample, and detrended with the average drift rate.
111
4. Time assistance from unsynchronised cellular networks
0 10 20 30 40 50 60 70 80 90
Time (h)
−40
−30
−20
−10
0
10
20
30
40
Ti
m
e
dr
if
t(
ns
/s
)
(a) BTSs 1a, 1b, 1c, and 1d
0 10 20 30 40 50 60 70 80 90
Time (h)
−600
−580
−560
−540
−520
−500
−480
−460
−440
Ti
m
e
dr
if
t(
ns
/s
)
(b) BTS 2
0 10 20 30 40 50 60 70 80 90
Time (h)
−40
−30
−20
−10
0
10
20
30
40
50
Ti
m
e
dr
if
t(
ns
/s
)
(c) BTS 3
0 10 20 30 40 50 60 70 80 90
Time (h)
−40
−30
−20
−10
0
10
20
Ti
m
e
dr
if
t(
ns
/s
)
(d) BTS 4
0 10 20 30 40 50 60 70 80 90
Time (h)
−40
−30
−20
−10
0
10
20
30
Ti
m
e
dr
if
t(
ns
/s
)
(e) BTS 5
0 10 20 30 40 50 60 70 80 90
Time (h)
−50
−40
−30
−20
−10
0
10
20
Ti
m
e
dr
if
t(
ns
/s
)
(f) BTS 6
Figure 4.3: Drift rates of the BTS clocks estimated from the synchronisation bursts.
Results have been smoothed.
112
4.4. Time stability of the GSM air interface
10−1 100 101
Time interval (h)
10−11
10−10
10−9
A
lla
n
de
vi
at
io
n
(s
)
(a) BTSs 1a, 1b, 1c, and 1d
10−1 100 101
Time interval (h)
10−10
10−9
10−8
A
lla
n
de
vi
at
io
n
(s
)
(b) BTS 2
10−1 100 101
Time interval (h)
10−11
10−10
10−9
A
lla
n
de
vi
at
io
n
(s
)
(c) BTS 3
10−1 100 101
Time interval (h)
10−11
10−10
10−9
A
lla
n
de
vi
at
io
n
(s
)
(d) BTS 4
10−1 100 101
Time interval (h)
10−11
10−10
10−9
10−8
A
lla
n
de
vi
at
io
n
(s
)
(e) BTS 5
10−1 100 101
Time interval (h)
10−10
10−9
A
lla
n
de
vi
at
io
n
(s
)
(f) BTS 6
Figure 4.4: Allan standard deviations of the times of arrival of the synchronisation
bursts.
113
4. Time assistance from unsynchronised cellular networks
Table 4.3: Time offsets and drift rates of the synchronisation bursts.
BTS Time offset (s) Average drift rate (ns/s)
1a 7741.826 −0.01
1b 10 554.790 −0.01
1c 10 125.746 0.00
1d 11 501.398 0.00
2 3064.425 −491.30
3 11 345.324 0.08
4 6962.260 0.25
5 6972.263 −0.02
6 2943.406 0.00
Figure 4.2 shows the estimated times of transmissions of the synchronisation bursts.
The TOT, in this case, was a direct indication of the time stability of the clocks because
the receiver was static. Some of the base stations were not in the line-of-sight of the
receiving antenna, although the effects of multipath are too small to be visible at the
scale of the figure.
We note there is no general behaviour of the time deviations shown in Figure 4.2.
The plots of the estimated drift rates, in Figure 4.3, are noisy but do not suggest any
common behaviour of the drift rates either. I did not want to increase the level of
filtering of the drift rate estimates to avoid smoothing too much the results and hiding
short-term instabilities. BTSs 1a, 1b, 1c, 1d, and 3 are the most stable and their time
deviations evoke the random-walk behaviour of a free-running oscillator. On closer
examination, BTS 1d shows some rapid oscillations that may indicate a system is
controlling the oscillator. It is possible that the sampling period of the measurement is
too slow to highlight the operation of a feedback mechanism on BTSs 1a-d. Besides,
the Allan standard deviation of BTSs 1a-d indicates a stability better than 1 × 10−10
over 10 hours. The maximum observation period of the plot of the sample Allan
standard deviation is limited to about 10 hours and there is no trend indicating that
the long-term oscillator stability worsens for longer observation periods. The good
stability properties and the presence of deterministic oscillations suggest the oscillator
is not free-running but is disciplined by an external reference oscillator. BTSs 1a-d
have a common behaviour and may be controlled by the same mechanism, possibly
located at the BSC. I was not able to locate BTSs 1a-d with certainty although a simple
triangulation experiment suggested that they were spread between two different sites.
Their frequencies are allocated to the same network operator, suggesting that they may
share the same BSC or that their equipment is provided by the same manufacturer.
114
4.4. Time stability of the GSM air interface
The time deviations of BTS 2, 4, 5, and 6 have sudden changes that can also be
evidence of external feedback mechanisms controlling the oscillators. The most striking
example is the BTS 5, shown in Figure 4.2(e). The time deviations of that BTS have
a regular pattern of seven oscillations of about one-hour period and an amplitude
of 10µs, which repeats with a period of about 24 h. This may be caused by the daily
calibration of the clock with an external frequency reference. Signals from the other
BTSs did not show such periodicity.
The offsets and the drift rates of the synchronisation signals, which are used as
observations of the BTS clocks, are shown in Table 4.3, after correction for ageing of the
frequency standard. The time offsets presented in Table 4.3 are calculated in reference
to the origin of the current hyper-frame and are not corrected for the signal propagation
time. We can see that none of the base stations has the same time offset. We might
have expected that all the BTS belonging to the same cell or attached to the same BSC
had synchronised hyper-frames. This is however not the case; we do not observe any
synchronisation of the hyper-frame origins between any base station. It is possible
that the feedback mechanisms noticed previously only control the frequencies of the
oscillators and that the global synchronisation of the network is never considered.
Therefore, it is likely that all BTS oscillators were disciplined by external mechanisms
that provided feedback about the frequency errors of the oscillators. However, as
implied by the time offset in Table 4.3, the absolute counters of the clocks did not seem
to be synchronised to any common reference.
Figure 4.4 shows the Allan standard deviation of the time of arrival of the synchro-
nisation bursts. The Allan standard deviation is used outside its domain of validity,
since most of the time deviations do not seem to come from a free-running oscillator.
Nevertheless, it provides some useful information about the long-term stability of each
BTS. We can see that the synchronisation signals of most base stations have a stability
better than 1 ppb over 10 hours. BTS 2 is the only base station for which even the short
term stability is worse than 10 ppb, although it is still within the 50 ppb limit set by the
3GPP standard.
Favey et al. (2001) also noted that the BTS clocks were disciplined by a common
oscillator. They recorded GSM times-of-arrival for 12 hours and observed a time
deviation of less than 10µs on one network and less than 1.5µs on another. They
showed that a feedback mechanism was operating on one network by inserting time
step changes of about 0.92µs, which corresponds to the quarter of a GSM symbol
period.
115
4. Time assistance from unsynchronised cellular networks
4.5 Prediction of time deviations
The time stabilities of the synchronisation bursts are not what we would expect of free
running oscillators. Feedback mechanisms introduce large changes at random times.
Yet, as we have seen in the previous section, the long-term time stabilities of the base
station oscillators are better than 1 ppb over 24 hours in most circumstances. The aim
of this section is to evaluate whether a simple filter, based on a model of a free-running
oscillator, can predict the time deviations from measurements taken several hours
beforehand. I reused the measurements from the experiment presented in Section 4.4.
System model and oscillator parameters
Most base stations, with the exclusion of BTSs 1a-1d, are not free-running oscillators
and thus cannot be modelled using simple linear models with Gaussian statistics.
Nevertheless, I investigated the prediction performance of a linear Kalman filter with
such models.
I assume that the measurement model includes an additive white Gaussian noise.
This noise model may not be strictly accurate and Faragher et al. (2010) found that
the distribution of the TOA error was better described by a Cauchy distribution than
a Gaussian distribution. However, the Gaussian noise assumption provided a good
starting point for the analysis. I investigated two possible values for the standard
deviation of the noise: a fixed value and the maximum-likelihood estimate based on
the measurements. The fixed standard deviation of the measurement noise, σz, was set
to 0.5µs, or about 150 m, as measured by Faragher (2007). The maximum likelihood
estimate of the measurement noise is described in the next section.
I considered two types of oscillator process models. I first modelled the oscillator
using the two-state model described by Equation 4.1. The filter thus tracked the time
deviation, bi(t), and the drift rate,
.
bi(t), of each base station i. I also considered the
simpler and more generic model of a random walk, where only the time deviations bi(t)
were tracked.
The maximum-likelihood estimates of the h-parameters were calculated by numer-
ical optimization of the log-likelihood function of the Kalman filter, as described in
Chapter 2.10. This was in an off-line method, which was applied only after a large
number of samples had been collected. The filter was then run again with the previous
measurements and with the optimised covariance matrix. On-line methods, for instance
the adaptive Kalman filter introduced by Mehra (1970), are also possible: they perform
both the state estimation and the covariance parameter optimisation using all the
116
4.5. Prediction of time deviations
measurements made. I chose off-line estimation for its simplicity and because it was
closer to what a practical system would deploy. Such a system may include a network
server to collect information from several devices and perform the optimisation. Up-
dated oscillator parameters are then communicated to the devices, thus avoiding the
need for the device to perform a computationally-demanding operation. If network
communication is not available, the device may use an on-line method, with degraded
performance.
Typical h-parameters lie in the range of 10−30 to 10−10 and they are scale parameters
for the likelihood function. I improved the conditioning of the optimisation problem by
using the logarithm of the h-parameters, rescaled and shifted to fit the appropriate range
of the optimisation algorithm. I used the Broyden-Fletcher-Goldfarb-Shanno (BFGS)
algorithm provided by the Scipy library (see Jones et al., 2001) to perform the numerical
optimisation. The values of the maximum-likelihood parameters are shown in Table 4.4
with an optimised measurement noise.
We can look at the optimised h−2 parameters to analyse the long-term stabilities of
the signals. The signals from base stations 1a-1d are the most stable and those from
BTS 2 are the least stable, regardless of the optimisation of the measurement noise.
Signals from BTS 3 have the second best long-term stabilities when the measurement
noise is optimised (in Table 4.4), but signals from BTS 5 and 6 have better stabilities
when the measurement noise is fixed. BTS 3 is the only BTS with an estimate of the
measurement noise larger than 0.2µs of standard deviation and it is 0.07µs larger than
the second-largest measurement noise. We can corroborate these estimates by observing
that the measurements in Figure 4.2(c) are noisier than the others. The reasons for a
large measurement noise are probably a weaker signal strength and a more important
multipath propagation for this BTS.
Table 4.4: Maximum-likelihood h-parameters of the process model and of the
standard deviation of the measurement noise σz
BTS σz (µs) h0 (s) h−2 (s−1)
1a 0.15 1.6 × 10−17 4.3 × 10−30
1b 0.13 9.1 × 10−17 8.9 × 10−32
1c 0.16 3.4 × 10−17 3.1 × 10−32
1d 0.16 1.2 × 10−16 2.0 × 10−32
2 0.11 6.6 × 10−15 2.6 × 10−15
3 0.25 4.0 × 10−16 3.5 × 10−28
4 0.12 4.1 × 10−17 2.0 × 10−25
5 0.15 3.0 × 10−22 4.1 × 10−22
6 0.14 9.6 × 10−17 1.2 × 10−24
117
4. Time assistance from unsynchronised cellular networks
The model can be checked using the normed innovation squared (NIS) of the Kalman
filter, which was introduced in Section 2.1. The NIS follows a χ2 distribution with two
degrees of freedom if the filter is properly “tuned” to produce an optimal estimate
from the measurements. The NIS is not a universal measure of the goodness-of-fit of
the model to the process being studied. More advanced methods may be required
to compare different models. However, the NIS provides a useful and simple way
to check the goodness-of-fit of the process and measurements models by comparing
the sample distribution of the NIS with its theoretical distribution. We visualise the
comparison between the sample and the theoretical distributions on probability plots,
also called Q-Q plots.
Figure 4.5 shows the probability plots of the NIS when the model parameters are
tuned to a random-walk model and Figure 4.6 shows the probability plots of the NIS
for a the two-state oscillator model. There is generally a good agreement between the
theoretical and observed distributions of the NIS for both models, except for BTS 2.
This is expected since we saw in Section 4.4 that the timing signals from BTS 2 have the
largest dynamics and seem to be derived from a disciplined clock with large corrections.
There is no significant difference between the one-state and two-state models. The
NIS tends to be biased in different directions for BTS 4, which may result in different
errors. The oscillations observed in the signals from BTS 5 do not appear on the NIS
distributions. This is probably because the diffusion and h-parameters are too large to
consider the oscillations as process noise.
I tried an ad-hoc solution which consisted of “artificially” increasing the state
covariance matrix every time the NIS crossed a threshold, for example six times the
normal standard deviation. This technique is sometimes called innovation gating in
the Kalman filter literature. However, this method did not significantly improve the
prediction performance. The prediction interval may have included more than one step
change of the frequency drift, which could not have been modelled by simply increasing
the process covariance. The control mechanisms that introduced the large frequency
drift variations appeared to be triggered at random times, which hindered long-term
predictions. Even more sophisticated approaches, such as the interacting Markov
model Kalman (IMM) filter (Blom et al., 1988) or the switching Kalman filter (Murphy,
1998), may not have improved the prediction performance because the process that
created the frequency drift variations could not be modelled accurately.
The results are shown with a maximum-likelihood estimate of the measurement
noise variance. The measurement noise depends on the position of the user and
thus can not be known or distributed in a practical system. I thus chose a constant
118
4.5. Prediction of time deviations
and conservative value of the measurement noise variance in the rest of this chapter,
compatible with the existing literature about the measurement noise of propagation
times in urban environments, and I chose σz = 0.5µs.
0 1 2 3 4 5 6
Theoretical quantiles
0
1
2
3
4
5
6
O
bs
er
ve
d
qu
an
ti
le
s
(a) BTSs 1a, 1b, 1c, and 1d
0 1 2 3 4 5 6
Theoretical quantiles
0
1
2
3
4
5
6
O
bs
er
ve
d
qu
an
ti
le
s
(b) BTS 2
0 1 2 3 4 5 6
Theoretical quantiles
0
1
2
3
4
5
6
O
bs
er
ve
d
qu
an
ti
le
s
(c) BTS 3
0 1 2 3 4 5 6
Theoretical quantiles
0
1
2
3
4
5
6
O
bs
er
ve
d
qu
an
ti
le
s
(d) BTS 4
0 1 2 3 4 5 6
Theoretical quantiles
0
1
2
3
4
5
6
O
bs
er
ve
d
qu
an
ti
le
s
(e) BTS 5
0 1 2 3 4 5 6
Theoretical quantiles
0
1
2
3
4
5
6
O
bs
er
ve
d
qu
an
ti
le
s
(f) BTS 6
Figure 4.5: Q-Q plots of the normalised innovation squared of the Kalman filter
with the one-state model.
Time prediction
I predicted the time deviations of the clocks using the same experiments as the ones
presented in Section 4.4. The predictions were performed using the one-state and the
119
4. Time assistance from unsynchronised cellular networks
0 1 2 3 4 5 6
Theoretical quantiles
0
1
2
3
4
5
6
O
bs
er
ve
d
qu
an
ti
le
s
(a) BTSs 1a, 1b, 1c, and 1d
0 1 2 3 4 5 6
Theoretical quantiles
0
1
2
3
4
5
6
O
bs
er
ve
d
qu
an
ti
le
s
(b) BTS 2
0 1 2 3 4 5 6
Theoretical quantiles
0
1
2
3
4
5
6
O
bs
er
ve
d
qu
an
ti
le
s
(c) BTS 3
0 1 2 3 4 5 6
Theoretical quantiles
0
1
2
3
4
5
6
O
bs
er
ve
d
qu
an
ti
le
s
(d) BTS 4
0 1 2 3 4 5 6
Theoretical quantiles
0
1
2
3
4
5
6
O
bs
er
ve
d
qu
an
ti
le
s
(e) BTS 5
0 1 2 3 4 5 6
Theoretical quantiles
0
1
2
3
4
5
6
O
bs
er
ve
d
qu
an
ti
le
s
(f) BTS 6
Figure 4.6: Q-Q plots of the normalised innovation squared of the Kalman filter
with the two-state model.
120
4.5. Prediction of time deviations
two-state linear models of the oscillators, which were tuned to the maximum-likelihood
parameters calculated in the previous section. A Kalman filter was used to estimate
the time deviations at a given time and then to propagate the state to predict the
time deviations at a future time. The prediction interval was varied between one and
twenty-four hours and the GPS time was observed for about ten minutes at every
update. The filter was initialised with an estimate of the clock offset and drift rate from
the first 100 measurements. I then calculated the errors between the predicted values
and a linear interpolation of the nearest measurements.
The prediction errors using the one-state oscillator model are shown in Figure 4.7.
The one-state model is a random walk, hence it can not track the frequency drift of an
oscillator. The prediction step consists of simply reporting the current time deviations
at a future time. This method is similar to assuming a constant error in the time
deviations, which could be done with a solution simpler than a Kalman filter. However,
the Kalman filter may reduce the prediction error since it may improve the estimation
of the current time deviations. These time deviations can be predicted with errors better
than 50µs for all base stations, except BTS 2. BTS 4 has the second-worst prediction
performance and it has a prediction error bias of about 30µs for prediction intervals
longer than 12 h. The normalised innovation squared of the model, shown in Figure 4.5,
suggests the model is biased when using the measurements from BTS 4. A similar bias
is observed in the NIS plots and the prediction errors of BTSs 1a-d and 3, although the
magnitudes of the biases are smaller than for BTS 4. BTS 4 has the largest amplitudes of
the time fluctuations of all the BTSs except BTS 2, as shown in Figure 4.2, hence it has
the largest bias, since the one-state model simply propagates the current time deviation.
The results using the two-state oscillator model are shown in Figure 4.8. The time
prediction errors are generally worse than in the case of the one-state model. It is
interesting to note that the NIS plots, in Figure 4.6, do not suggest such a large difference
between the one-state and two-state models.
The time prediction errors are mostly caused by changes in the clock drift rates,
probably triggered by control feedback mechanisms. The two-state oscillator model
does not bring any advantage over the simple random-walk model, which in fact gives
the smallest prediction errors. Tracking the drift rate component of the oscillator, as
done in the two-state model, is not worthwhile when considering controlled oscillators.
The dynamic linear model assumes that the frequency drift follows a random walk,
although a disciplined oscillator generally controls the frequency with discrete step
changes. We could improve the filter by adding a discrete state variable, which estimates
if the clock is in feedback mode, although such a model seems impossible to design
121
4. Time assistance from unsynchronised cellular networks
1 4 8 12 16 20 24
Time ahead (h)
−4
−2
0
2
4
Er
ro
r
qu
an
ti
le
(µ
s)
(a) BTSs 1a, 1b, 1c, and 1d
1 4 8 12 16 20 24
Time ahead (h)
−30000
−20000
−10000
0
10000
20000
30000
Er
ro
r
qu
an
ti
le
(µ
s)
(b) BTS 2
1 4 8 12 16 20 24
Time ahead (h)
−10
−5
0
5
10
Er
ro
r
qu
an
ti
le
(µ
s)
(c) BTS 3
1 4 8 12 16 20 24
Time ahead (h)
−40
−30
−20
−10
0
10
20
30
40
Er
ro
r
qu
an
ti
le
(µ
s)
(d) BTS 4
1 4 8 12 16 20 24
Time ahead (h)
−10
−5
0
5
10
Er
ro
r
qu
an
ti
le
(µ
s)
(e) BTS 5
1 4 8 12 16 20 24
Time ahead (h)
−15
−10
−5
0
5
10
15
Er
ro
r
qu
an
ti
le
(µ
s)
(f) BTS 6
Figure 4.7: Box-plot of the prediction errors of the GSM time signals, using a one-
state oscillator model (random walk model). The red horizontal line is the median
value; the blue central rectangle extends from the 25-percentile to 75-percentile;
the black horizontal lines mark the 10-percentile and 90-percentile; and the blue
crosses indicate the maximum and minimum values.
because the control mechanism does not seem to follow any predictable law.
122
4.6. Tracking of local oscillator bias
1 4 8 12 16 20 24
Time ahead (h)
−4
−2
0
2
4
Er
ro
r
qu
an
ti
le
(µ
s)
(a) BTSs 1a, 1b, 1c, and 1d
1 4 8 12 16 20 24
Time ahead (h)
−1000
−500
0
500
1000
Er
ro
r
qu
an
ti
le
(µ
s)
(b) BTS 2
1 4 8 12 16 20 24
Time ahead (h)
−500
0
500
Er
ro
r
qu
an
ti
le
(µ
s)
(c) BTS 3
1 4 8 12 16 20 24
Time ahead (h)
−30
−20
−10
0
10
20
30
Er
ro
r
qu
an
ti
le
(µ
s)
(d) BTS 4
1 4 8 12 16 20 24
Time ahead (h)
−40
−20
0
20
40
Er
ro
r
qu
an
ti
le
(µ
s)
(e) BTS 5
1 4 8 12 16 20 24
Time ahead (h)
−2000
−1000
0
1000
2000
Er
ro
r
qu
an
ti
le
(µ
s)
(f) BTS 6
Figure 4.8: Box-plot of the prediction errors of the GSM time signals, using a
two-state oscillator model.
4.6 Tracking of local oscillator bias
We now exploit the long-term stability of the BTS clocks to correct the time offset of the
local oscillator of a mobile device. We consider a practical application similar to the
one described in the previous section: a mobile device tracks the time deviations of
several reference clocks with the goal to provide fine-time assistance to a GPS receiver,
thus improving the TTFF. In the previous section, we considered the time deviations
of the BTS clocks individually and we did not use the short-term stability of the local
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4. Time assistance from unsynchronised cellular networks
oscillator. This model is now improved by building a clock ensemble, which contains all
the clocks in the system, including the local oscillator.
Ensemble clock filter
Time is measured by a clock whose process is inherently noisy. We can filter the
observations from a clock to improve the time estimate if the noise characteristics are
known, as was done in the previous section. We can also improve the time estimation
by combining the measurements from several clocks, since the noise processes are
generally uncorrelated between different clocks. A filter typically keeps track of the
time fluctuations of each clock with respect to a common time reference. A better
estimate of the time reference is then obtained by a weighted average of the filter
estimates.
The case of UTC raises specific problems since the goal is to define a stable time
scale and not simply to track the deviations of several clocks. The UTC time scale is
estimated by an ensemble of atomic clocks from several laboratories. This system is
not fully observable: the time scale is undetermined and only time differences between
the atomic clocks are observed. This leads to convergence issues with the ensemble
filter and Galleani et al. (2003) reviewed some of the possible solutions. The same
problem occurred when the GPS time scale was established. An early approach relied
on a master clock, which served as the reference time of all time observations. The
GPS composite clock, presented by Brown (1991), was then developed to build an
independent time scale. However, we do not have such a problem when estimating the
local time. All the time measurements in a mobile receiver have a common time scale
and the clock ensemble considered in this section is thus fully observable. The time
deviations of all the clocks are measured with respect to the GPS composite clock given
by the GPS software receiver, albeit with measurement errors.
I evaluated the performance of the ensemble filter for the two process models
presented in Section 4.3. I only describe the equations of the clock ensemble filter
for the two-state oscillator model, as the equations of the random-walk model are
straightforward to derive. The state vector of the clock ensemble with the two-state
process model is
x(t) =
[
blocal(t)
.
blocal(t) b1(t)
.
b1(t) · · · bN(t)
.
bN(t)
]T
,
where blocal(t) is the time deviation of the local oscillator and b1...N(t) are the time
deviations of the synchronisation bursts from the base stations. All times are in the GPS
124
4.6. Tracking of local oscillator bias
system time scale. The state vector is estimated by a Kalman filter and the estimate is a
Gaussian random variable with mean xˆ(t) and covariance matrix P(t).
The time of transmission of the synchronisation bursts from base station i, received
at time t in the receiver time scale, is noted t(GSM)i (t). The GPS time, calculated at time
t in the receiver time scale, is noted t(GPS)(t). The relationships between these time
measurements and the components of the state vector are given by
blocal(t) = t − t(GPS)(t)
bi(t) = t
(GSM)
i (t) − t(GPS)(t) for i = 1 . . .N .
There are two measurement models, depending on whether a GPS measurement
is available or not. Most of the time, the filter tracks the time deviations of the BTS
clocks using the times of arrival of the synchronisation bursts. These measurements are
performed in the local time scale of the receiver whereas the state vector is expressed
in the GPS time scale. The difference between the local and the GPS time scales are
not known until a GPS measurement is made, hence the system is not fully observable
using the GSM timing signals only. If a GPS observation is available, the time deviations
can be directly observed
zi(t) =
[
blocal(t) bi(t)
]T
+ R w(t),
with the corresponding observation matrix being a 2× 2N matrix and H0,0 = 1, H1,2i = 1,
and all the other elements set to 0. The observation noise covariance matrix is
R =
σ2GPS 00 σ2GSM + σ2GPS,
 .
where σGSM and σGPS are the standard deviations of the GSM and GPS measurement
noises, respectively.
If a GPS measurement is not available, the time deviation of the BTS i is measured
with respect to the local oscillator and the measurement equation becomes
zi(t) =
[
t(GSM)i (t) − t
]T
+ R w(t)
= [bi(t) − blocal(t)]T + R w(t).
The corresponding observation matrix is a 1 × 2N matrix with H0,0 = −1, H0,2i = 1,
and all the other elements set to 0. The observation noise covariance matrix is simply
R =
[
σ2GSM
]
.
We remark that the observation time, t, is both a measurement and a variable of
the process model. We assumed a noise-free measurement of the local time, although
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4. Time assistance from unsynchronised cellular networks
in theory there may be some noise if the triggering of the GSM or GPS measurements
is not perfectly synchronised with the local clock. In practice, I did not observe such
problems.
Local oscillator tracking experiment
The experiment is representative of a mobile device with a low-cost local oscillator
and with access to both GPS and GSM signals. The goal is to track the time bias of
the device’s clock in order to provide fine-time aiding to a GPS receiver, whenever the
user needs it. At periodic intervals, the mobile device performs a GPS acquisition and
calculates its position and the bias of its local clock in reference to the GPS time. The
device also receives the synchronisation signals from the GSM base stations and feeds
them to a clock ensemble filter. The latter, provided with a model of the BTS oscillators
and the local clock instabilities, keeps an estimate of the local clock bias between the
GPS updates. The GSM base stations provide time measurements that have worse
stabilities and accuracies than the time derived from GPS signals, but they are typically
stronger than GPS signals and can be acquired indoors.
The experimental arrangement, outlined in Figure 4.1, was similar to the one used for
the experiment presented in Section 4.4. The only change was the source of the triggering
signal, which was derived from a mass-market Quartz crystal oscillator instead of
the more-accurate atomic frequency reference standard. The trigger signal was used
both as the local clock and as the signal that synchronised the time measurements.
This arrangement highlighted that the sampling rate did not have to be constant. The
sampling instant t was only required to synchronise the measurements of the local
clock, t(local)(t), and the GSM time signals, t(GSM)(t). The role of the local oscillator of
the mobile device was performed by the crystal oscillator embedded in the digital
counter card. Its performance was unknown and the datasheet only mentioned its
frequency accuracy, of 50 ppm over the temperature range of −20 ◦C to 70 ◦C. I opened
the acquisition board and identified the oscillator as a Pletronics CMOS Quartz crystal
oscillator, without temperature compensation.
The GPS signal was recorded only once per hour to limit the storage space require-
ments. We can determine the time bias of the local clock accurately only during these
GPS measurements, which is not problematic for the filter operation since the drift
rate is estimated over long observation periods. However, the filter performance was
evaluated by calculating the error between the estimated time bias and the observed
time bias, which was available only during the GPS measurements. We thus need
an accurate estimation of the local time bias at a higher rate than the GPS updates in
126
4.6. Tracking of local oscillator bias
Rb
IFR
Input 2
Input 1
Counter
TriggerExt Osc
Nordnav-RX
GPIB
NI interface
TCP/IP
PC 1
Network
PC 2
RPC
5 s
1 MHz
GPS RF
GSM RF
XO
TCXO
storage
Figure 4.9: Schematic diagram of the experimental arrangement using a Quartz
oscillator. The diagram does not represent the real physical location of the
components.
order to evaluate the tracking performance of the filter for every GSM measurement.
I interpolated the bias of the local clock between the GPS measurement periods. I
first tried classical interpolation functions, such as the linear and spline models, but
these methods created a large tracking error of the filter between the interpolation
points, at the GPS update times. I considered that the origin of these errors was
not physical but in fact a poor fit of the interpolation functions. I suspect that the
random-walk behaviour of an oscillator cannot be modelled with enough accuracy
by interpolation functions with such low polynomial orders. A Gaussian process
regression, as described by Rasmussen (2006, chapter 2), resulted in smaller residual
errors than the other interpolation methods. I therefore used a Gaussian process, with
a squared-exponential covariance matrix of characteristic length four hours, in order to
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4. Time assistance from unsynchronised cellular networks
provide the time bias of the local clock with respect to the GPS time.
The ensemble filter was initialised with the average bias and the average drift rate
of each clock, estimated over the first three hours of data collection. In a practical
application, the filter could be initialised from estimates kept in a central database or
transmitted from nearby devices. The process covariances of the BTS clocks were set to
their maximum-likelihood estimates, calculated in Section 4.5.
Most mobile devices already keep an estimate of the current ageing of the local
oscillator but do not optimise the noise covariance parameters. We could calculate
the maximum-likelihood parameters of the local clock from the less noisy GPS time
measurements. However, it is necessary to keep a history of all previous measurements
to run the filter, which may not be practical in a real application. Instead, I tried a few
combinations of h-parameters within two orders of magnitude of the parameters of
a typical low-quality oscillator. I kept the parameters that gave the smallest tracking
error: h0 = 10−16 and h−2 = 10−17.
I set the cross-covariance terms to zeros, thus making the assumption there was no
correlation between any two clocks of the ensemble. This is not entirely correct since
we expect that some BTSs are connected to a common clock reference and that local
propagation effects may introduce some correlation between the signals arrival times. I
also ran the filter with the cross-correlation terms set to the sample cross-covariance
calculated from the measurements taken in Section 4.4 and the changes in the results
were less than 1µs.
GPS-aiding of the local oscillator
0 10 20 30 40 50 60 70 80 90
Time (h)
−1.6
−1.4
−1.2
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
O
bs
er
ve
d
ti
m
e
er
ro
r
(m
s)
Figure 4.10: Time bias of the local clock, recovered from GPS time measurements.
Figure 4.10 shows the error in the time calculated by the local clock, using the GPS
interpolation as true time.
128
4.6.
Tracking
oflocaloscillator
bias
0 10 20 30 40 50 60 70 80 90
Time (h)
−5
0
5
10
T
i
m
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
(
µ
s
)
(a) 12-hour GPS updates.
0 10 20 30 40 50 60 70 80 90
Time (h)
−4
−2
0
2
4
6
8
T
i
m
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
(
µ
s
)
(b) 48-hour GPS updates.
0 10 20 30 40 50 60 70 80 90
Time (h)
−2
0
2
4
6
8
10
T
i
m
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
(
µ
s
)
(c) No GPS update after initialization.
Figure 4.11: Time series of the local time error after filtering, all base stations,
one-state model.
0 10 20 30 40 50 60 70 80 90
Time (h)
−2
0
2
4
6
8
T
i
m
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
(
µ
s
)
(a) 12-hour GPS updates.
0 10 20 30 40 50 60 70 80 90
Time (h)
−2
0
2
4
6
8
T
i
m
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
(
µ
s
)
(b) 48-hour GPS updates.
0 10 20 30 40 50 60 70 80 90
Time (h)
−2
0
2
4
6
T
i
m
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
(
µ
s
)
(c) No GPS update after initialization.
Figure 4.12: Time series of the local time error after filtering, all base stations,
two-state model.
129
4.
T
im
e
a
ssista
n
c
e
fr
o
m
u
n
sy
n
c
h
r
o
n
ise
d
c
e
l
l
u
l
a
r
n
e
t
w
o
r
k
s
0 10 20 30 40 50 60 70 80 90
Time (h)
−20
−10
0
10
20
30
T
i
m
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
(
µ
s
)
(a) 12-hour GPS updates.
0 10 20 30 40 50 60 70 80 90
Time (h)
−10
−5
0
5
10
15
20
T
i
m
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
(
µ
s
)
(b) 48-hour GPS updates.
0 10 20 30 40 50 60 70 80 90
Time (h)
−10
0
10
20
30
40
50
T
i
m
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
(
µ
s
)
(c) No GPS update after initialization.
Figure 4.13: Time series of the local time error after filtering, least stable base
stations, one-state model.
0 10 20 30 40 50 60 70 80 90
Time (h)
0
10
20
30
40
T
i
m
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
(
µ
s
)
(a) 12-hour GPS updates.
0 10 20 30 40 50 60 70 80 90
Time (h)
0
20
40
60
80
T
i
m
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
(
µ
s
)
(b) 48-hour GPS updates.
0 10 20 30 40 50 60 70 80 90
Time (h)
0
20
40
60
80
100
120
140
160
T
i
m
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
(
µ
s
)
(c) No GPS update after initialization.
Figure 4.14: Time series of the local time error after filtering, least stable base
stations, two-state model.
130
4.7. Discussion and conclusions
I analysed the results in four different scenarios. I first ran the filter with the
measurements from all the nine BTSs available. The tracking errors are reported in
Figure 4.11 for the one-state model and in Figure 4.12 for the two-state model.
We are also interested in the performance of the filter in absence of stable BTSs
signals. I ran the filter again with all the measurements except those from BTSs 1a to 1d
and BTS 3, which were the most stable BTSs. This was to test the behaviour of the filter
when accurate measurements were not included in the ensemble. The tracking errors
are shown in Figure 4.13 with the one-state model and in Figure 4.14 with the two-state
oscillator model.
As expected, the performance varied depending on the stabilities of the signals and
on the number of states in the process models. The two-state model performed slightly
better than the one-state model when the signals from all BTSs were included. The
maximum time error with a two-state model, shown in Figure 4.12, was about 8µs and
the median time error was under 1.2µs, for all GPS update periods. The maximum
error with a one-state model was 10µs and the median error remained about the same,
except with no GPS update where the median error of the one-state model was about
3µs.
The absence of the most stable BTS signals showed a different picture, illustrated in
Figures 4.13 and 4.14. The results were significantly worse than when all the BTSs were
included. This was expected since the ensemble filter weighed the time measurements
according to their time stabilities. The lack of stable measurements affected the estimate
of the ensemble, as the weights of the different sources of measurements were now
approximately similar. The two-state model performed significantly worse. The
two-state filter wrongly estimated the drift rate of some of the BTS signals, leading to
an increasing time error that reached more than 70µs after 24 h. The maximum error,
without any GPS update, was about 160µs after three and a half days. It is interesting
to note than the estimation error was always positive, which may be explained by a bias
in the initial drift rate estimate. However, even a time accuracy of 160µs can benefit a
GPS L1 C/A receiver, whose code period is 1 ms.
4.7 Discussion and conclusions
GPS time estimation using fractional pseudo-ranges
The software receiver I used to calculate the GPS time in this chapter extracts the full
pseudo-range measurements of the GPS signals; however, most of the receivers found
in mobile devices only use fractional pseudo-ranges, as explained in Section 2.5. The
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4. Time assistance from unsynchronised cellular networks
time ambiguity of fractional pseudo-range measurements is an integer number of the
PN code period. Syrjärinne (2000) compared simulations and experimental results of
the recovery of GPS time from a millisecond-ambiguity algorithm. He concluded that
the GPS time could be accurately recovered under strong signal conditions and a good
constellation geometry. However, the time error may be too large for sub-millisecond
accuracy in degraded situations. His simulation results showed that the GPS time could
be calculated with an accuracy of 3 ms, with an integer ambiguity of 1 ms. The results
he obtained from a GPS receiver connected to a GPS simulator were significantly less
accurate, with errors up to 50 ms in poor constellation geometry and the millisecond
ambiguity was not evident.
Recovering the GPS time from fractional measurements is difficult because the
times of transmission of the GNSS signals are unknown and need to be estimated by
the algorithm. A receiver that decodes the navigation signals can accurately determine
the TOT from the time-of-week word in the navigation message. Otherwise, algorithms
based on fractional measurements must estimate the TOT, as explained in Section 2.5.
An error in the estimation of the time of transmission results in an error in the GPS
time estimate and in the user position. A position error r introduces a TOT error of
r/r˙, where r˙ is the radial velocity, also called range rate, of the satellite seen from the
user position. The timing error, t, is proportional to the propagation speed of the
signal, which is approximately c. We therefore have t = r/r˙. GPS satellites have radial
velocities of at most 600 m/s and the pseudo-range measurement errors are typically
30 m. The error in the GPS time estimation is thus about 50 ms in an assisted-GPS
receiver.
Assisted-GNSS receivers can thus have difficulties in estimating accurately the local
clock bias. This inaccuracy is not critical in mass-market receivers, since the user is
often only interested in its position. However, the clock ensemble filter we analysed
in this chapter requires accurate time measurements in order to provide an estimate
of the GPS system time with a sub-millisecond accuracy. I suspect that the estimation
of GPS time from fractional measurements is a problem only during the initialisation
of the ensemble filter. The iterative algorithms used to solve the position and time
equations are generally sensitive to the starting point, as highlighted by Syrjärinne
(2000). I expect the time errors to be negligible when fine-time aiding is available to
initialise the iterative algorithm with an uncertainty smaller than the PN code period.
An alternative method would be to make fractional measurements whose time
ambiguity is large, hence resulting in a smaller risk of large errors. Akopian et al. (2009)
propose the detection of the bit edges of the navigation signal, which has a period of
132
4.7. Discussion and conclusions
20 ms. The accuracy of the time, using simulated measurements, is negligible though
an integer ambiguity of 20 ms can appear at C/N0 lower than 30 dB−Hz. The new GNSS
signals have longer code periods, which improves the errors introduced by the time
ambiguities.
Receiver motion
All the experiments were performed with a static receiving antenna. The filter presented
in this section can not estimate the GPS system time if the receiver is moving, since it
can not distinguish whether the changes in the reception times of the bursts are caused
by changes in the transmission times or in the propagation times. Put differently, all
the GSM time measurements have an error caused by the confusion between the time
deviation of the oscillator and the motion of the receiver. The filter hence needs to
estimate the propagation time of the signals as well as the time deviations of the BTS
clocks.
One solution is to include the user position in the clock ensemble. The filter updates
the mobile position estimates whenever positioning measurements are available, for
example using a GNSS receiver. The propagation times can then be calculated, assuming
the positions of the BTSs are known, and removed from the times-of-transmission
measurements. However, the goal of the experiments presented in this chapter was
to provide precise time when GNSS signals were not available, for example indoors.
Indoor positioning is notoriously difficult, but the accuracy required for fine-time aiding
is of tens of microseconds, or about 3 km, which can be attained in most situations using
Cell-ID or RSS positioning. The maximum range within a GSM cell is typically 15 km,
or a maximum time error of 50µs, which is acceptable for the purpose of time assistance
of a GNSS receiver. In comparison, the maximum drift rate of the local oscillator in the
experiment was about 35µs/h. Solo Matrix, proposed by Duffett-Smith et al. (2004), is a
positioning method relying on GSM signals. It assumes short-term stabilities of the base
stations and calculates the mobile position using the time difference of arrival of GSM
signals whose time offsets have been calibrated. Solo Matrix could be easily improved
by an ensemble clock filter to propagate the time offsets of the BTSs, thus enabling a
more robust positioning from GSM signals than relying only on the short-term stability
of one BTS.
Another solution is to use the timing advance (TA) parameter of the GSM protocol,
which is proportional to the range between the mobile and its serving base station.
The base stations set the TA parameter to command the mobile stations to introduce
a time offset at the beginning of their transmissions. This delay ensures the signals
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4. Time assistance from unsynchronised cellular networks
transmitted by users at different distances arrive within their correct time slots at the
serving BTS. The granularity of the TA parameters is of one symbol for the round-trip
delay, i.e. about 3.7µs or a distance of 550 m. The TA is necessary because of the TDMA
approach used by GSM; it is always implemented and defined in 3GPP (2003b, TS
05.10). Some CDMA systems may also provide distance information as CDMA systems
need to adjust their transmission power in order to limit the near-far effect. The TA, or
its CDMA equivalent, is provided only by the serving BTS. The corrections are thus
limited to only one base station, which is a problem since the goal of the ensemble filter
is to exploit the diversity of the measurements from several base stations. However,
the receiver motion can be averaged out by using the measurements from several BTSs,
which are assumed to be uniformly distributed around the receiver.
Conclusions
The results presented in Section 4.6 show that the GPS time can be predicted within 30µs
using a single daily GPS measurement, for a static receiver. This stability is better than
1 ppb over 24 hours. Most oscillators, like TCXO, can achieve this level of stability over
short periods. However, the long-term stabilities of the BTS synchronisation signals
could be used to provide fine-time aiding to a GNSS receiver after a period of sleep.
The mobile device tracks the time deviations of the signals from the neighbouring BTSs
as long as GNSS time and positions can be calculated. The device can then be turned
off and still be able to obtain accurate GPS system time immediately after resuming
its operations. There is, however, a possibility of time error if the user has moved.
Fine-time aiding reduces the time-to-first-fix, thus providing a better experience to the
user and saving battery by limiting the time spent in the power-consuming acquisition
stage.
134
CHAPTER 5
GNSS aiding in synchronised
cellular networks
I discussed in the previous chapters the theoretical times-to-first-fix (TTFFs) of assisted-
GNSS receivers and how the TTFF can be reduced by providing the approximate GPS
time to the receiver. The experimental analysis of various GSM networks in Cambridge,
United Kingdom, showed that the time signals of these networks are not synchronised
and have variable levels of stabilities. Some networks can provide fine-time aiding
whereas others are not stable enough. Besides, the GSM synchronisation signals only
carry counters that are not related to the GPS time or any universal time scale. The GPS
system time can only be determined after having first calibrated the time origins of the
counters.
Synchronised networks, such as, for example, CDMA2000, can provide stable and
synchronised time information, which can help improve the TTFF performance of a
GNSS receiver. CDMA2000 is a technology of synchronised networks mostly deployed
in North America, Latin America, and Asia. The standard specifies that the networks
must be synchronised with the GPS system time, which opens new perspectives for
improving the GPS performance on mobile personal devices. GPS system time can
be recovered accurately from the CDMA2000 signals and then used to reduce the
GPS signal acquisition search space, thus improving the TTFF performance. The
synchronisation of the signals can also be exploited to perform radio positioning, using
either only the CDMA2000 signals or in combination with GPS signals. In this chapter,
we investigate how the features of synchronised networks can improve the performance
of positioning on mobile devices.
135
5. GNSS aiding in synchronised cellular networks
5.1 Introduction
CDMA2000 networks
CDMA2000 is a standard of third-generation cellular networks for personal communi-
cations. It is an evolution of IS-95, which is a standard of second-generation networks
developed by Qualcomm Inc. under the name cdmaOne™. Most of the specifications of
cdmaOne™ are proprietary. CDMA2000 is more accessible to the researcher since it is
recognized by the International Telecommunication Union (ITU) and, for the most part,
by the 3rd Generation Partnership Project 2. We only address CDMA2000 networks
in this chapter. However, the work presented in this chapter may only require the
specifications of an IS-95 network.
The air interface of the network is based on code-division multiple access with
synchronised transceivers. Furthermore, the standard 3GPP2 (2004a, C.S0002) specifies
that the time scale employed for the synchronisation of the receivers is the GPS time
scale. This architecture is different from the GSM and UMTS standards: networks
using GSM or W-CDMA, the most common air interface of UMTS, are unsynchronised
networks.
The time synchronisation requirements are specified by 3GPP2 (1999, C.S.0010),
although the tolerance value is not written explicitly in this document. The document
refers to the specifications of IS-95, which are not available in the public domain.
However, publicly-available information suggests that all base stations must be syn-
chronised within ±2µs of the GPS system time. In practice, this is achieved by using
GPS-controlled oscillators at the base stations. The synchronisation requirements can
be challenging to meet and they increase the operating costs of a CDMA2000 networks.
The networks also need to keep operating without a GPS signal and the standard
specifies that the synchronisation must be maintained within ±10µs of the GPS system
time for eight hours after the GPS signal becomes unavailable. I noticed that most base
stations had two GPS antennas, which may provide for redundancy of the time source
but also indicates that a dependable and accurate time source is critical for the reliable
operation of the network. Some major network providers in North America have
started to switch their CDMA2000 networks to UMTS with W-CDMA since 2012. The
Long-Term Evolution standard (3GPP - LTE) has been retained for the fourth-generation
networks in North America, and this standard is an evolution of UMTS.
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5.1. Introduction
Existing work
I previously discussed GNSS aiding in unsynchronised networks. Synchronised
networks have received more attention because their synchronisation and stability
can be exploited to determine the user’s position. The “Enhanced-911” mandate of
the FCC, discussed in Section 1.3, has fostered research in this area because the FCC
requirements could not easily be met using other solutions, for example using received
signal strength. Caffery et al. (1998a) present a comprehensive overview of positioning
methods in CDMA networks. They consider angle-of-arrival methods, which are
generally suitable only at the base stations, and time-of-arrival methods. They also
review the statistical properties of some time-of-arrival estimators.
I was not able to find relevant literature about the experimental analysis of the
time stabilities of CDMA2000 signals. It seems that most positioning methods based
on CDMA2000 signals assume perfect synchronisations and stabilities of the clocks
generating the signals. Little information is available from the industry about the
timing and synchronisation of CDMA2000 equipment.
All publications highlight the problem of position errors introduced by non-line-of-
sight (NLOS) signals and provide different mitigation solutions. Later in this chapter, I
will address the NLOS problem both at the TOA estimation and at the positioning levels.
The assumption is generally made in the existing literature that the user position is
tracked for at least several seconds or longer. Tracking improves the position accuracy
because it exploits the accumulation of measurements, but it may also leave the users
frustrated by the time required to determine their positions accurately. Algorithms
that are designed to provide an almost instantaneous position are generally called
“snapshot”, or “push-to-fix”. I only look at snapshot algorithms in this chapter.
Outline of the chapter
The work presented in this chapter is based on experiments on the synchronisation and
stability of a CDMA2000 network. I first investigated the time stability of a CDMA2000
network with an outdoor antenna. The synchronisation of the signals with the GPS
time scale allowed me to measure the propagation time of the signals. I analyse the
statistics of the propagation time in both an indoor and outdoor environments in this
chapter. I then consider a radio-positioning system based on the cellular signals. I
present its performance here and illustrate some algorithms to improve the robustness
and the availability of the position estimates. Finally, I exploit the synchronisation of
the cellular signals with GPS to develop a positioning system that combines the signals
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5. GNSS aiding in synchronised cellular networks
from both the GPS and the CDMA2000 network. I evaluated the performance of this
system on indoor measurements, using a high-sensitivity GPS acquisition method, and
I discuss the benefits of this approach at the end this chapter.
Acknowledgements
I carried out the experiments described in this chapter at the University of Calgary,
Canada, in the “Position, Location, And Navigation” group. I received assistance
from Professor Gérard Lachapelle, Dr Ali Broumandan, and Dr James T. Curran,
who provided the experimental apparatus and the database of the BTS positions and
identifiers.
5.2 Measurements of times of arrival of CDMA2000 signals
I carried out several experiments on the CDMA2000 network in different environments
and propagation conditions. The apparatus common to all these experiments is
presented in this section. Variations made to this baseline arrangement will be
explained when appropriate in the remaining sections.
Structure of CDMA2000 signals
CDMA2000 is a code-division multiple access system in which all the transceivers
of a network generally share the same frequency band. CDMA is provided by a
direct-sequence spread spectrum modulation of the signal. Each logical channel is
modulated by a different orthogonal spreading code so that all the logical channels can
be transmitted on the same physical channel. The radio-frequency carrier is generally
in a band around 800 MHz or 1900 MHz. The signal is spread twice on the forward link.
The logical channels are first spread by a pseudo-noise sequence based on a Walsh code.
Additional encoding, such as interleaving, convolutional error coding, etc. is then
applied to most of the logical channels. Finally, the baseband signal is spread by a pilot
PN sequence, whose generating polynomial is specified in 3GPP2 (2004a, §3.1.3.1.18).
The signal goes through a baseband filter whose impulse response is specified in 3GPP2
(2004a, Table 2.1.3.1.18.1-1), which limits the bandwidth to about 1.23 MHz.
The Forward Pilot Channel is transmitted by every base station in order to synchro-
nise the mobile receivers. The Forward Pilot Channel is assigned the Walsh function 0
and does not undergo any further processing: it is a direct modulation of the RF carrier
by the pilot PN sequence. The pilot PN sequence is a quadrature sequence composed of
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5.2. Measurements of times of arrival of CDMA2000 signals
215 chips transmitted at a chip rate of 1.2288 Mchips/s, resulting in a repetition period
of Tpilot = 80/3 = 26.66 . . . ms.
The base stations are identified by the offset index of the pilot PN sequence. The
origin of the pilot PN sequence is delayed by 64N chips, where N is the offset index that
identifies the base station. The granularity of 64 chips implies there is a maximum of
512 base stations visible from any point of the network. The transmission range of a base
station must also be limited so that the code-phase delay introduced by the propagation
time of the signal remains within the same PN offset index. The length of 64 chips
thus limits the maximum range to about 15.6 km. This limit must be enforced by the
network operator, even if the transmitting power of the BTS equipment is allowed a
greater range, otherwise interference between base stations may be introduced.
Some CDMA2000 channels may use faster spreading rates and wider bandwidths.
In particular, spreading rates of three times and five times are available on many
channels. I did not investigate these channels, as they are not explicitly designed to
provide time synchronisation and their availability is uncertain. However, their wider
bandwidths may be of interest for future work.
Previous work on time estimation in multipath channels
The bandwidth of the CDMA2000 synchronisation signal is not wide enough to
characterise the full impulse response of the channel, i.e. all the propagation paths
can not be resolved. Thus, it is not possible to resolve the arrival times of all of the
signals coming from different paths and the possibilities of multipath mitigation are
limited. Faragher (2007) showed that, by the effects of convolution of the channel
impulse response in a narrow bandwidth, the earliest correlation peak may actually
occur before the signal has arrived.
Most of the existing work on multipath mitigation does not apply directly to our
problem. Existing publications generally assume that filtering the measurement stream
over time is possible, whereas we are interested in “snapshot” positioning, where the
position is calculated with typically less than 500 ms worth of measurements. Fleury
et al. (1999) proposed to use the space-alternating generalized expectation-maximisation
(SAGE) algorithm to estimate the channel parameters. The parameters were the signal
amplitudes, the path delays and the directions of arrival of the signals. The direction of
arrival could be determined because they used an antenna array. The SAGE algorithm
was introduced by Fessler et al. (1994) as a method to find the maximum-likelihood
parameters of a probabilistic model. It is a variant of the expectation-maximisation
(EM) algorithm developed by Dempster et al. (1977), an iterative method to estimate
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5. GNSS aiding in synchronised cellular networks
the maximum-likelihood parameters of a likelihood function. The SAGE approach
improves the convergence speed for a class of problems and has been widely used for the
estimation of communication channels. The SAGE algorithm applied to TOA estimation
is similar to super-resolution methods, such as the multiple signal identification and
classification (MUSIC) algorithm and the estimation of signal parameters by rotational
invariance techniques (ESPRIT) algorithm. For example, MUSIC was used by Klukas
(1997) to improve the TOA estimation of signals on the Advanced Mobile Phone System.
Leon et al. (2006) and Li et al. (2009) compared the performance of MUSIC, ESPRIT, and
SAGE, and concluded that the SAGE algorithm gave the smallest mean squared error.
The SAGE algorithm can be used to reduce the effects of multipath propagation,
however this method has some limits. First, SAGE can be computationally demanding
and it is sensitive to the actual number of multipath components, which is generally
unknown and assumed to be fixed. Another problem is that the largest peak is not
necessarily the first arrival, since multipath can attenuate the line-of-sight component
below the strengths of the non-line-of-sight components at time. In this condition,
the LOS signal may fall below the detection threshold and the estimator will not find
the LOS signal. Finally, the SAGE estimator is still limited by the channel bandwidth,
which is not wide enough to allow for a fine resolution of the channel impulse response.
Signal model in presence of multipath propagation
The received signal at baseband is represented by z(t). We assume the channel has an
additive white Gaussian noise of dual-sided power spectral density N0. It is a multipath
channel composed of Np paths, with a flat-fading profile and is not frequency selective.
We note 1(t) the replica of the transmitted signal, 1(t) = cp(t) ∗ h(t), where cp(k) is the
pilot PN sequence and h(t) is the impulse response of the pulse-shaping filter. The
received signal, after baseband downconversion and sampling, is modelled by
z(tk) =
Np∑
l
al 1(tk − τl)e j2piδ f tk + σ ν(tk) (5.1)
where tk is the sampling instant, al is the complex amplitude of the l-th path, τl is the
code phase of the signal from the l-th path, and δ f is the residual frequency offset after
downconversion. We seek to estimate τl from the received samples. We define two
vectors of parameters, the vector of amplitudes, a = [a0, . . . , aNp] and of code phases,
τ = [τ0, . . . , τNp]. ν(tk) is a sequence of normal complex random variables such as
〈ν(t)ν∗(t′)〉 = 2δ(t − t′). The synchronisation parameters are the PN code phase and the
frequency offset.
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5.2. Measurements of times of arrival of CDMA2000 signals
The three estimators of the code phase parameter are discussed in the rest of
this section. The time-of-arrival estimate of the signal is calculated from the code-
phase estimate by removing the constant offset index used to identify the BTS. The
signal is detected using a generalised likelihood-ratio test (GLRT), as presented in
Chapter 2, with a coherent integration period of 3Tpilot = 80 ms. This choice represents
a compromise between longer integration times to improve the detection probability
and shorter integration times to reduce the sensitivity to frequency offset mismatch.
Besides, an integration time that is an integer number of periods of the GPS C/A code
facilitates the synchronisation of the measurements with GPS time. The recommended
maximum Doppler shift to limit the losses during the coherent integration is 6.25 Hz,
corresponding to a maximum velocity of 3.5 km/h at the RF frequency of the CDMA2000
network. I was careful to walk slowly enough while making the experiments.
The stabilities of the oscillators at the receiver and the BTSs were such that I did not
have to adjust the frequency offset even for measurements made three weeks apart.
All the base stations of a same network provider transmitted on the same frequency,
thus the frequencies offsets of all the signals received by a static receiver were the same.
Both the receiver and the transmitting BTSs were static hence the frequency offset δ f
was determined once for all the base stations and for the duration of the experiment.
The frequency offset was removed from the search space, which then consisted only of
the possible PN pilot code phases.
Three methods to estimate the times of arrival
I used three different estimators of the times of arrival for the calculations presented in
the rest of this chapter:
• the single-path, maximum-likelihood estimate, which is the same as the code-
phase estimation method based on the generalised likelihood-ratio test presented
in Chapter 2, and is the maximum-likelihood estimator under the single-path
assumption, i.e. it is optimal only if Np = 1;
• a heuristic estimator, which is similar to the code-phase estimator except that it
gives more than one estimate, in an attempt to estimate the multipath components.
The estimated parameters is the set, Θˆ, of all the code phases that give a local
maximum of the likelihood ratio above the detection threshold, η. This set is
more formally defined as
Θˆ =
{
∀ τ ∈ [0,Tpilot) |Λ(τ) > Λ(τ − Ts) and Λ(τ) > Λ(τ + Ts) and Λ(τ) > η
}
,
where Λ(τ) is the likelihood ratio of the signal, defined in Equation 2.12; and
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5. GNSS aiding in synchronised cellular networks
• a SAGE-based estimator, which employs the SAGE estimator to calculate the
maximum-likelihood values of the amplitudes and delays parameters. The
likelihood function of the signal for the amplitude and code-phase parameters is
logL(a, τ|z1:N) ∝ Ez − 2 Re
 N∑
k=1
zk
Np∑
i=0
a∗i1
∗ (tk − τi)
 ,
where N is the coherent integration length, and z1:N is a vector of N samples of
the received signal. The parameters estimated by SAGE are thus
(aˆ, τˆ) = arg max
(a,τ)
L(a, τ|z1:N).
A decision rule keeps the earliest code-phase estimate in the delay vector whose
associated amplitude crosses the detection threshold η.
Choosing the number of paths is a well-known problem when modelling a prop-
agation channel. Different techniques have been investigated, for example Wax et al.
(1985) used the minimum description length as an information criterion. Most of these
methods are computationally expensive since they iterate the SAGE algorithm over
different values of Np. My implementation of the SAGE algorithm did not estimate
the number of paths. I calculated the value of the maximum likelihood for different
number of paths for the signals coming from BTS 440 and over a sample of outdoor
measurements. The maximum likelihood was obtained for three paths and I kept this
value for the rest of the analysis.
SAGE is an iterative algorithm and I used a stopping rule based on a fixed threshold
on the residual delay and amplitude steps. I found that few iterations were required.
For many base stations, the estimates were almost unchanged after the first iteration.
In these situations, the algorithm is similar to the successive-interference-cancellation
(SIC) scheme,as described by Patel et al. (1994). In many situations, the SIC scheme
may provide results similar to SAGE while being less computationally demanding.
Experimental details
I collected the data in the laboratory of the “Position, Location, And Navigation”
group at the University of Calgary, Canada. The experimental arrangement was
built around a National Instrument (NI) radio-frequency vector signal analyser with
three channels. Each channel consisted of a NI PXI-5600 RF receiver, followed by a
NI PXI-5142 signal digitizer. The samples were streamed directly onto an array of
15 hard-disk drives, thus enabling the high data transfer rates required to record the
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5.2. Measurements of times of arrival of CDMA2000 signals
signals continuously acquired by the three channels. I configured the instrument for a
10 MHz input bandwidth and a 10 MHz baseband sampling rate. The recorded data
were the in-phase and quadrature samples of the baseband signal.
Different sets of antennas were used. For outdoor measurements, I used the
GPS and CDMA2000 antennas permanently installed on the roof of the “Centre for
Communications and Information Technologies” building. For indoor measurements, I
used a patch antenna for GPS and a quarter-wavelength dipole for CDMA2000 signals.
The CDMA2000 antenna was followed by a ZHL-1217 low-noise amplifier with a gain
of about 30 dB and noise figure of 1.5 dB, and an RF pre-amplifier of gain 20 dB and
noise figure 5 dB. Low-noise amplifiers with gains of about 23 dB were integrated to
the GPS antennas. A pre-amplifier with a gain of 20 dB was also added to the GPS
antenna for indoor experiments. The three acquisition channels were synchronised
together with common clock and trigger signals. One of NI PXI-5600 RF receivers
provided the clock signal from an OCXO with an accuracy of 50 ppb and a stability
of 20 ppb/year. The lengths of the cables to the roof-top antennas were unknown and I
assumed there was no significant length difference between the signal paths of the GPS
and the CDMA2000 channels.
All the measurements I made are divided into two series in this chapter. The
outdoor signals were recorded as sequences of duration 30 seconds each, repeated
every 24 minutes for 90 hours. The interval between acquisitions was longer than the
one used for the GSM experiments presented in Chapter 4 because the sampling rate
of the NI frontend was higher, thus filling the available storage space more quickly. I
did not reduce the sampling rate or decimate the signal in order to exploit the widest
bandwidth possible to study the channel propagation characteristics. I recorded this
data set during the Thanksgiving week-end in November 2011. There was little activity
in the buildings surrounding the receiver. This area of the city was composed mostly of
two-story houses, with taller buildings on the university campus and a nearby hospital.
The building was located in a valley surrounded with low-height hills. I would
subjectively classify the outdoor propagation characteristics as those of a moderate
urban environment.
The set of indoor measurements was smaller, consisting of eight sequences lasting
37.5 seconds each. The first six sequences were collected on the 21st November 2011 and
the last two sequences on the 29th November 2011. Each sequence corresponded to a
different location of the indoor antenna in the laboratory room. In some sequences, the
antenna was moved to represent the motion of a pedestrian. I will make reference to the
configuration of a specific sequence when necessary. One of the acquisition channels
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5. GNSS aiding in synchronised cellular networks
was always connected to the GPS outdoor signal for all experiments. I extracted the
GPS time from this signal and used it to compensate for the drift of the signal analyser’s
oscillator.
I did not observe any near-far effect, which often degrades CDMA signals. The
near-far effect is caused by the difference in signal strength between the nearest and
furthest transmitters: the dynamic range of the receiver may not be large enough to
detect the weakest signal. This effect is specific to CDMA communications, where all
the signals continuously occupy the same spectral band. The NI PXI-5600 frontend
had a spurious-free dynamic range of 80 dB and the NI PXI-5142 had a 14-bit ADC, or
about 84 dB of dynamic range. The dynamic range of the receiver was thus about 80 dB,
which accommodated the difference of signal strength between the strongest signal,
from BTS 232, and the weakest one, from BTS 292. The near-far effect is problematic in
mass-market receivers, whose dynamic ranges are smaller than the apparatus used
for the experiments. The CDMA2000 standard helps mitigate the near-far effect by
continuously adapting the power of the signals transmitted by the mobile stations, so
that the noise floor is not significantly increased in the area around the transmitter.
Another solution is the virtual cumulative blanking method proposed by Duffett-Smith
et al. (2002).
5.3 Time stability and synchronisation of CDMA2000
networks
The CDMA2000 standard specifies that the pilot signal must be synchronised within
±2µs of the GPS time, but the stability of the synchronisation is not specified. An
oscillator of the type seen in Chapter 4, for example, may present quasi-periodic
instabilities within ±2µs and still satisfy the synchronisation requirement. There
does not seem to be any publicly-available information about the synchronisation
characteristics of real networks. Therefore I measured the time stabilities of the pilot
signals of seven CDMA2000 base stations.
Signal strength
The experimental arrangement was similar to the one described in Section 5.2, using
outdoor antennas. The time-of-arrival errors were calculated using the distances
between the BTS positions in the database and the surveyed position of the CDMA2000
antenna at the receiver.
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5.3. Time stability and synchronisation of CDMA2000 networks
The post-detection signal-to-noise power ratio is shown in Table 5.1. The SNR
is estimated from a simple mean-variance estimator, which may be biased and not
statistically robust. However, the results give an idea of the range of signal strengths.
There was no significant multipath fading in the outdoor signals. The receiver and the
transmitter were all static, although the presence of several roads with dense traffic
could have provided some dynamic scatterers near the receiver.
Table 5.1: Post-integration signal-to-noise ratio.
BTS SNR (dB)
80 19.8
164 15.0
180 9.4
232 39.8
292 6.1
356 10.0
440 29.1
Time synchronisation
I calculated the time differences between the theoretical and observed times of arrival
of the pilot signals. The theoretical time of arrival was calculated from the GPS time,
the PN offset of the base station, and the distance between the receiver and the BTS.
The observed times of arrival were estimated by the SAGE algorithm.
Table 5.2: Pilot signal TOA error, theoretical - measured.
BTS Median error (µs) Standard deviation (µs)
80 −0.33 0.13
164 0.08 0.05
180 0.27 0.12
232 1.98 0.04
292 −1.63 0.15
356 −16.47 0.20
440 1.66 0.05
The statistics of the universal time errors of the pilot signals are presented in
Table 5.2. All of them are within the synchronisation requirement of 2µs, except for
BTS 356.
BTS 356 was located on one of the tallest building in the city centre and thus had
longer range than most base stations. It is possible that this BTS had a visible range
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5. GNSS aiding in synchronised cellular networks
larger than the 15 km required to maintain the maximum code delay within a same
offset index of the BTS. A delay of 16µs, about 5 km, may have been introduced to
apparently move the BTS away from its interference zone. This possibility is reinforced
by the fact that the PN offset index of this BTS was never reused in the entire city,
contrary to most of the other BTSs. Another possibility is that there was an error in the
BTS database, although this would imply that the base station was even further than it
appeared, which is unlikely. BTS 232 was close to the 2µs limit, although there was a
wider margin when I did not use the SAGE algorithm.
The time synchronisation errors can not be explained away by a difference in the
path lengths of the GPS and CDMA2000 signals at the transmitters, introduced for
example by different cable lengths. I noticed that the GPS antennas of most BTSs
were situated close to the equipment room, at the bottom of the mast supporting the
CDMA2000 antennas at its top. But the errors observed are one order of magnitude
larger the errors that could be introduced by the mast heights, which look at most 30 m
high. We will consider, in the rest of this chapter, that the values presented in Table 5.2
are the constant biases of the BTS clocks and we will use these values to correct the
TOAs when needed.
Time stability
We have studied the average time synchronisation of the network and we now focus of
the time stability of the signals. The time stability was analysed using the single-path
estimator. I did not apply a detection threshold for these measurements as the goal
was to utilise as many data points as possible. This introduced some outliers for the
BTSs with the weakest signals, since the maximum correlation peak may have been
caused by the noise alone. I applied a Kalman smoother to remove these outliers and I
excluded BTS 292, which had the weakest signal, from this rest of the experiments in
this section.
The Kalman smoother used a simple random-walk model and an innovation
gate that rejected measurements outside a 3σ normalised innovation squared. The
measurement noise is a combination of the uncertainty in the TOA estimation and
the uncertainty in the GPS time estimation. The sample standard deviation of the
TOA measurements is shown in Table 5.2 and I assumed that it was representative of
the measurement noise, i.e. that the network was perfectly stable. I chose 0.5µs as a
conservative value for the standard deviation of the measurement noise. The standard
deviation of the Gaussian random walk was arbitrarily set to 1 ns/s, a value typical of
OCXO oscillators. I gave the process noise a rather large variance intentionally, since
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5.3. Time stability and synchronisation of CDMA2000 networks
the smoother was simply used to remove the outliers. The goal was to give the most
flexibility to the process model.
The smoothed time series of the TOA errors are presented in Figure 5.1. The stability
of the pilot signal was excellent. It looked as if the BTS oscillators were always locked
on the GPS time and had excellent long-term stabilities. I observed some short-term
fluctuations, some of which periodic and with a period close to 24 h. These may have
been caused by daily changes in the temperature. The local time and the drift of the
local clock were estimated at every acquisition using the GPS data, so the changes in
temperature of the receiver should not have affected the results. The amplitudes of
these variations were smaller than 200 ns for all the base stations.
0 10 20 30 40 50 60 70 80 90
Time (h)
−0.20
−0.15
−0.10
−0.05
0.00
0.05
0.10
0.15
0.20
TO
A
er
ro
r
(µ
s)
Figure 5.1: Time series of the TOA errors, smoothed by a Kalman filter using a
random walk process model. The TOA has been offset to have a null error at the
time origin. BTS 292 is excluded as it generally falls below the detection threshold.
The cause of the remaining time deviations is difficult to establish. The GPS
software receiver I designed employed a delay-locked loop with a spacing of 100 ns
and a tracking bandwidth of 20 Hz. The C/N0 of the signal was large, generally about
48 dB−Hz, hence the noise introduced by the GPS time estimation was negligible. (For
a more complete treatment of performance bounds in tracking loops, one can refer to
Parkinson et al. (1996, chapter 7)). The two most likely sources of fluctuations are thus
the stochastic nature of the BTS oscillators and the TOA estimation errors, in particular
caused by propagation in an urban environment.
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5. GNSS aiding in synchronised cellular networks
Conclusions
The times of arrival the CDMA2000 pilot signals of the seven base stations studied
are within 2µs of GPS time, except for one base station, and the time stability is better
than 200 ns over a period of 90 hours. This suggests the CDMA2000 transmitters have
stabilities similar to those of the GPS master clock and are continuously synchronised
with the GPS time, with a constant clock bias error. The BTSs probably used disciplined
oscillators continuously locked onto the GPS signals.
5.4 Error analysis of the propagation time
We now make the assumption that the CDMA2000 pilot signals are perfectly synchro-
nised with GPS time, up to a constant clock bias. This assumption may not be entirely
correct as we noted some periodic variations which may indicate that the signals are
affected by temperature changes. However, it enables us to calculate the propagation
times of the signals with only one receiver. Experiments to study the propagation times,
such as those carried out by Faragher et al. (2010), generally require two synchronised
receivers. One receiver is located close the transmitter, at a place where the signal prop-
agation is little affected by the environment. The second receiver, synchronised with
the first, makes the measurements at the locations of interest. By using a CDMA2000
receiver synchronised with GPS time we can calculate the propagation times with only
one receiver, since the CDMA2000 transmitters are also synchronised with GPS time.
In this section, I present an analysis of the estimation errors of two time-of-arrival
estimators, both in an indoor and outdoor environments.
Sources of errors
TOA measurements are affected by different types of uncertainties. Some of these
uncertainties are noise processes that can be estimated reasonably well. These include,
for example, the phase noise and the sampling noise. Others, such as the ones caused
by the uncertainties and randomness of the propagation channel, are more difficult to
analyse.
The oscillators in the signal analyser and at at the base stations introduce some errors
in the TOA measurements. The long-term drift of the local oscillator is compensated
using the GPS measurements and I assume that the short-term fluctuations of the
TCXO in the RF frontend are negligible. The time deviations of the BTS oscillators
are not directly observable. I will assume that they are negligible, although we noted
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5.4. Error analysis of the propagation time
quasi-periodic time variations for one of the base station. The sampling period is 10 ns
and the analog-digital converter has more than 80 dB of dynamic range, hence the
quantisation noise can also be neglected.
The statistics of the TOA estimation error have been calculated for an additive white
Gaussian noise channel, without multipath, in Section 2.1. The estimation errors in
a multipath channel are more difficult to estimate and are part of an active research
area. Many analytical models of propagation channels have been proposed and most
of them are derived from physical considerations backed by empirical measurements.
They sometimes only address the calculation of the signal strength or the signal fading
properties in a multipath environment, and the statistics of the individual paths are not
available. The Nakagami model (Nakagami, 1960) is commonly used to represent a
fading environment and several models have been proposed to predict the average
signal strength and the capacity of mobile communication systems. These do not
attempt to model the propagation times.
We are more interested in the timing models of multipath propagation. Many
models of multipath propagation depend on parameters that are specific to a particular
environment, such as the number of paths, the position of the reflectors and scatterers,
etc. Saleh et al. (1987) developed one of the most commonly used models in indoor
environments, generally known as the Saleh-Valenzuela model. They present the power
profiles of measurements in an indoor channel and note that the signal arrivals form
clusters. They develop an analytical model where the arrival of a cluster and of the
signals within a cluster are both modelled as Poisson processes. The arrival times thus
follow an exponential distribution. The Saleh-Valenzuela model, as most analytical
models, is generally used for simulation and evaluation purposes, and not directly for
improving the propagation time estimation. We will consider this model for the indoor
measurements of propagation times of CDMA2000 signals.
Experimental details
The experimental arrangement was similar to the one described in Section 5.2. I
considered two types of estimators: the SAGE and the single-path maximum-likelihood
estimators. The TOA errors were calculated by removing the offset indices of the BTSs,
the constant time biases shown in Table 5.2, and the theoretical propagation times.
The previous section did not address the detection problem; all the measurements
were used by the Kalman smoother, and a gating process based on the likelihood
ratio of each sample was used to reject outliers. I added a likelihood-ratio test for the
remaining experiments of this chapter. I chose a detection threshold of η = 4, which
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5. GNSS aiding in synchronised cellular networks
gave a probability of false alarm of about 0.13, using Equation 2.15. I did not perform
any additional non-coherent integration. The same threshold was applied to each of
the three estimators.
Analysis of outdoor signals
The distribution of the TOA errors, estimated using SAGE, is shown in Figure 5.2. The
shapes of the histograms of the TOA noise of BTS 232 and BTS 440, in Figures 5.2(d)
and 5.2(f), suggest the time distribution has thicker tails towards late arrival times. The
probability plots of BTS 80, BTS 180, and BTS 356, show that the early arrival times are
more frequent that those predicted by a Gaussian arrival time distribution, e.g. longer
tails for early arrivals.
150
5.4.
Error
analysis
ofthe
propagation
tim
e
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
F
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−4 −3 −2 −1 0 1 2 3 4
Gaussian TOA quantiles
−4
−3
−2
−1
0
1
2
3
4
T
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(a) BTS 80
−0.4 −0.2 0.0 0.2 0.4
Observed TOA error (µs)
0
1
2
3
4
5
6
7
8
F
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−4 −3 −2 −1 0 1 2 3 4
Gaussian TOA quantiles
−4
−3
−2
−1
0
1
2
3
4
T
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s
(b) BTS 164
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0
1
2
3
4
F
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−4 −3 −2 −1 0 1 2 3 4
Gaussian TOA quantiles
−4
−3
−2
−1
0
1
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4
T
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(c) BTS 180
−0.4 −0.2 0.0 0.2 0.4
Observed TOA error (µs)
0
2
4
6
8
F
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−4 −3 −2 −1 0 1 2 3 4
Gaussian TOA quantiles
−4
−3
−2
−1
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1
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4
T
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(d) BTS 232
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
F
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−4 −3 −2 −1 0 1 2 3 4
Gaussian TOA quantiles
−4
−3
−2
−1
0
1
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T
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(e) BTS 356
−0.4 −0.2 0.0 0.2 0.4
Observed TOA error (µs)
0
2
4
6
8
F
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q
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n
c
y
−4 −3 −2 −1 0 1 2 3 4
Gaussian TOA quantiles
−4
−3
−2
−1
0
1
2
3
4
T
O
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r
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s
(f) BTS 440
Figure 5.2: Time-of-arrival errors, using SAGE-based peak detector.
151
5. GNSS aiding in synchronised cellular networks
We first calculate the CRLB of the TOA estimator, using Equation 2.8. I calculated
the effective channel bandwidth of the CDMA2000 Forward Pilot Channel to about
β = 1.1 MHz, based on the pulse-shaping filter response. The theoretical standard
deviation and the observed standard deviation, which are recalled from Table 5.2, are
presented in Table 5.3. The values are generally four times larger than the minimum
standard deviation of the estimator, with significantly worse standard deviation for
BTS 80, BTS 232, and BTS 440. The difference may be attributed to the propagation
channel or the stability of the oscillators, as mentioned in Section 5.4.
Table 5.3: Comparison of the Cramér-Rao bound of the TOA estimator and the
observed standard deviation.
BTS SNR (dB) Lower Observed
bound (ns) deviation (ns)
BTS 80 19.8 dB 11 135
BTS 164 15.0 dB 18 49
BTS 180 9.4 dB 35 78
BTS 232 39.8 dB 1 38
BTS 356 10.0 dB 32 112
BTS 440 29.1 dB 4 44
The TOA errors of the outdoor signals fit a Gaussian distribution reasonably well
in the range of ±2σ, where σ is the sample standard deviation. The distribution of
the estimated TOA errors is however more skewed towards early arrival times for
the BTS 80, BTS 180, and BTS 356; negative errors of magnitude larger than two
standard deviations are more exceptional than for the Gaussian fit. This is probably a
consequence of the SAGE estimation algorithm, which privileges early arrival times.
This anomaly is not observed on the other BTSs, where, on the contrary, early arrival
times have stronger probabilities than the Gaussian fit. This too may be explained by
the behaviour of the SAGE algorithm: we perform the TOA estimation in a multipath
environment under the assumption of a constant number of paths for all base stations.
BTS 232 and BTS 440 have the strongest signals and the presence of secondary paths
is clearly visible in the histograms of the TOA. The SAGE algorithm thus correctly
estimates the parameters of these secondary paths, which gives less freedom to identify
earlier arrival times. The secondary paths were not visible in the measurements from
the other BTSs, thus giving more opportunities for the algorithm to identify early paths.
We do not observe a significant skew towards later arrival times, as would be the case
in a multipath environment where the arrival times of the secondary paths follow a
Poisson process.
152
5.4. Error analysis of the propagation time
Analysis of indoor signals
The distributions of the TOA errors of the pilot signals in an indoor environment are
shown in Figure 5.3, using the SAGE algorithm, and in Figure 5.4 using the single-path
maximum-likelihood estimator.
153
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G
N
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a
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in
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ise
d
c
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l
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l
a
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n
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t
w
o
r
k
s
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
F
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Gaussian TOA quantiles
−4
−3
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(a) BTS 80
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
F
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(b) BTS 164
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
F
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−4 −3 −2 −1 0 1 2 3 4
Gaussian TOA quantiles
−4
−3
−2
−1
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(c) BTS 180
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
F
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−4 −3 −2 −1 0 1 2 3 4
Gaussian TOA quantiles
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(d) BTS 232
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.5
1.0
1.5
2.0
F
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−4 −3 −2 −1 0 1 2 3 4
Gaussian TOA quantiles
−4
−3
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−1
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(e) BTS 356
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
F
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−4 −3 −2 −1 0 1 2 3 4
Gaussian TOA quantiles
−4
−3
−2
−1
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T
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s
(f) BTS 440
Figure 5.3: Time-of-arrival errors, using SAGE-based peak detector.
154
5.4.
Error
analysis
ofthe
propagation
tim
e
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
F
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−4 −3 −2 −1 0 1 2 3 4
Gaussian TOA quantiles
−4
−3
−2
−1
0
1
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3
4
T
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(a) BTS 80
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
F
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(b) BTS 164
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
F
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(c) BTS 180
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
F
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(d) BTS 232
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.5
1.0
1.5
F
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(e) BTS 356
−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0
Observed TOA error (µs)
0.0
0.2
0.4
0.6
0.8
1.0
F
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(f) BTS 440
Figure 5.4: Time-of-arrival errors, using correlation peak maximum.
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5. GNSS aiding in synchronised cellular networks
Unsurprisingly, the error distributions of the times of arrival of indoor signals are
more complex than in the case of outdoor signals. There is no major difference between
the performance of the SAGE algorithm, shown in Figure 5.3, and the single-path ML
algorithms, shown in Figure 5.4. The Gaussian fit is generally poor, except for BTS 80 and
BTS 164. I did not calculate the standard deviation because such descriptive statistics
would not be meaningful without a better characterisation of the error distribution. The
TOA error distributions are generally biased towards late arrival times, which indicates
that even the SAGE algorithm is not capable of identifying reliably the shortest path
components in these signals. However, the SAGE algorithm does reduce the occurrence
of some early TOA errors. The worst estimation errors occur with signals from BTS 356
and BTS 440.
0 5 10 15 20 25 30
Time (s)
0
2
4
6
8
10
TO
A
de
vi
at
io
n
(µ
s)
Figure 5.5: Illustration of multipath fading, power profile of BTS 440.
The SAGE algorithm has a limited effect on the TOA estimation of BTS 440 because
the signals from that BTS can propagate along two paths with almost equal probabilities.
The strengths of these two components are almost similar, hence the histogram of
the single-path estimator (Figure 5.4(f)) has two modes of approximatively equal
frequencies. Figure 5.5 shows the power profile of the fading caused by the multipath
propagation of the channel. In this example, the receiver was static and the correlation
time of the multipath fading was several integration periods, generally between 400 ms
and 1.2 s. The signal fading of the two main paths is not an independent process,
since a fade occurring on one path tends to corresponds to a stronger signal along
this other path, thus keeping the total energy constant. The SAGE algorithm often
fails to recover the earliest arrival time as the strength of the early ray is below the
detection threshold. However, the magnitude of the secondary peak in the histogram
156
5.5. Positioning using cellular signals only
of Figure 5.3(f) is smaller than in Figure 5.4(f), thus highlighting the advantage of the
SAGE-based estimator.
The case of BTS 356 is interesting, as it shows a systematic bias of 1.2µs which
can not be corrected by any algorithm. This BTS is the tallest one and this may lead
to a greater probability of being in sight of a reflector that may provide a strong
non-line-of-sight signal to the receiver. It is possible that most of the techniques based
on maximisation of the likelihood of a single signal fail in this situation, since the
likelihood function is generally maximised by the strongest ray. A joint likelihood
function, which includes the signals from all the base stations and a model of biases,
may help in this situation. This scheme will be discussed later in this chapter.
5.5 Positioning using cellular signals only
We now exploit the synchronisation of the BTSs to determine the position of a mobile
receiver. This method is similar to the Advanced Forward-Link Trilateration (AFLT),
specified in the standard 3GPP2 (2004c, C.S0022-A). I have not been able to find
any experimental evaluation of AFLT or any report about the usage of AFLT in real
networks. I present the results of indoor and outdoor experiments.
Positioning in an urban environment or indoors is generally challenging because
most of the signals propagate in multipath conditions and the TOA measurements are
often made of non line-of-sight signals. The TOA estimators presented in the previous
section cannot completely eliminate the effects of multipath propagation and NLOS
conditions. For example, Figures 5.3(e) and 5.3(f) show that the times-of-arrival of the
signals from BTS 440 and BTS 356, using the SAGE estimator, are incorrect because
of non line-of-sight propagation. We review and evaluate two algorithms that help
mitigate the effects of non line-of-sight propagation.
Positioning with NLOS signals
The analysis of the TOA error distribution, in Section 5.4, has highlighted the effects
of multipath propagation. The indoor signals are all non line-of-sight and suffer from
systematic delays, such as the signals from BTS 356, or delays varying between two
discrete values, for instance the signals from BTS 440. Multipath propagation and
non-line-of-sight measurements are generally found together. The term multipath is
often associated to the problems of channel and TOA estimations, whereas NLOS is
more commonly found when talking about position calculation.
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5. GNSS aiding in synchronised cellular networks
The robustness of the positioning algorithm can be improved in order to cope
with NLOS signals. Robust statistics is a research area that deals with improving the
performance of an estimator when the input data break the assumptions made by the
estimator. The TOA estimated from non-line-of-sight signals break the Gaussian and
line-of-sight assumptions of the algorithm used for positioning. The effects are variable,
from a systematic bias of a hundreds of meters, to the divergence of the iterative
procedure and, in the worst case, the convergence to a grossly incorrect solution. We
now review several robust positioning methods that have been proposed to mitigate
NLOS conditions.
A first class of methods seeks to classify whether each measurement belongs to a LOS
or NLOS signal, before any position is calculated. Fischer et al. (1998), presenting the
results of simulations, address the detection of NLOS measurements by comparing the
sample variances of the pseudo-range measurements. Measurements whose variances
are larger than the average variance are treated as NLOS. Wylie-Green et al. (2001)
consider a channel with intermittent NLOS conditions. A set of measurements is
collected for every transmitted signal and they calculate the probabilities of LOS of
NLOS for every measurement. These probabilities are propagated by a transition model,
which depends on the speed of the mobile, and updated by the measurement likelihood.
The calculations are performed over sub-sets during which the user is assumed to be in
the same area, such that the user motion is masked by the measurement noise. The
TOAs of the LOS measurements are described by Gaussian random variables with
fixed means, while the TOAs of the NLOS measurements have an additional positive
random variable. They propose an algorithm that propagates the probabilities of LOS
and NLOS of each measurement and present the results of simulations.
A different approach consists in using the results of the position calculation to
detect the NLOS components in the original measurements. Pi-Chun Chen (1999)
proposes the calculation of the least-squares positions for all possible combinations
of measurements, then take the average position weighted by the reciprocal of the
residuals. This approach is computationally demanding as it requires to solve the
position equation
∑N
i=3
(N
i
)
times for a 2D-position, where N is the number of input
signals. It improves the robustness of the iterative least-squares method but it does
not specifically address NLOS signals, ignoring, for example, the property that NLOS
signals are more likely to arrive later than the LOS signals.
The two categories of methods we have presented above try to address the problem
at two different stages: some methods work on the measurements while others work
on the results of the calculation. It is possible to combine these methods in order to
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5.5. Positioning using cellular signals only
embed the probabilities of LOS and NLOS into the position calculation. Cong et al.
(2004) investigate this approach and they compare two algorithms. The first algorithm
successively rejects the combinations of measurements based on their residuals. The
second algorithm takes a probabilistic approach, which relies on prior probabilities
of LOS and NLOS of every measurement. The joint probability density function of
the position is then calculated. The measurement likelihood function of the LOS path
is a Gaussian model, and a combination of Gaussian model and exponential arrival
time for the NLOS path. The joint PDF can be multimodal, in particular if several
NLOS measurements exist and their prior probabilities are not known accurately.
Further measurements are used to update the PDF and the position is estimated
from the maximum a posteriori. Their approach has the advantage of including
probabilistic information about the LOS/NLOS states of the signals. They highlight
that the performance, in a simulation environment, depends on the number of detected
signals.
All these publications evaluate the proposed methods in simulations. Kim et al.
(2001) provide experimental results of a position method in an urban environment,
using CDMA signals. They use a two-stage Kalman filter: a first filter estimates the
signal biases and a second filter calculates the position after correction for the biases.
The authors report errors of about 300 m at the beginning of the experiment and under
50 m after 3 min of tracking.
Two methods to mitigate non-line-of-sight effects
We consider the user of a mobile device, such as a smartphone, who wants to know
his/her horizontal position as quickly as possible, even if the device has just been
turned on. This approach is sometimes called “snapshot” or “push-to-fix” in contrast to
position tracking, for which the measurements are accumulated and the user position
is continuously tracked. We set the maximum latency of the position calculation to
500 ms.
The device makes the measurements and calculates the position, without assistance
from network equipment. One of the methods presented by Caffery et al. (1998a) requires
angle-of-arrival information from the base stations, which may only be possible with
assistance from the network. It is also possible that the position is calculated on a
network server and may then be transmitted back to the user. This approach is taken
by some of the methods developed for the E-911 mandate, when the computing power
of the user device is too limited.
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5. GNSS aiding in synchronised cellular networks
The position is estimated from the iterative least-squares algorithm presented in
Chapter 2. The position equation is simpler than for GPS signals: the transmitters are
static, hence calculating the time of transmission of the signals is not required and a
two-dimensional position can be determined from at least three measurements. The
code period of the pilot signal, Tpilot, is longer than the GPS L1 C/A signal and this
almost guarantees the convergence of the algorithm if the initial position is known
within 8000 km and the dilution of precision is reasonable. We assume the pseudorange
measurements have additive white Gaussian noise of standard deviation 60 m, and
the measurements are independent between BTSs. The iterative algorithm is stopped
when the square root of the sum of the squared residuals becomes smaller than 0.1 m.
The initial position is set within 1.5 km of the true position, with could be provided
using RSS or Cell-ID positioning for instance.
Most BTSs only cover a sector of about 120° and they are grouped by three, with
different identifiers, to cover a cell. I ignored the angular information about the user
position contained in the coverage of the BTSs, although it could improve the position
accuracy by constraining the solution inside a cellular sector.
Most of the algorithms for NLOS mitigation are not able to work within the
snapshot scenario we are interested in, as they require measurements collected over
longer periods of time. We investigate two of the methods that do not require such
filtering:
• the residual rejection algorithm is an ad-hoc method mentioned in Cong et al.
(2004), among others. It consists of iteratively calculating the position, rejecting
the measurement whose residual is the largest and above a given threshold, and
repeating until all the measurement residuals are below the rejection threshold. I
chose a threshold of 3σ, where σ is the standard deviation of the measurement
noise; and
• the full enumeration of all the combinations of measurements, developed by
Pi-Chun Chen (1999).
Some algorithms, such as the one designed by Wylie-Green et al. (2001), estimate the
probability of a signal being a LOS, NLOS, or a false alarm. I believe this method is the
most appropriate for positioning with NLOS signals, since it embeds the uncertainty on
the propagation models. However, the classification of a signal can only be performed
with enough measurements and a good prior position, which are not available in a
snapshot scenario.
The full enumeration is particularly computationally demanding and is unlikely
to be practical on a smartphone. It is, however, presented for completeness, since we
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5.5. Positioning using cellular signals only
expect that it provides better results. Experiments will show that this method is actually
inefficient and inaccurate.
We wish to give more weight to early signals when dealing with a set of TOA
candidates from a same BTS, as in the all-peaks detector. I found that a useful heuristic
was to change the variance of the measurement noise as a function of the order of
arrivals of the detected paths. I used a simple linear function, where the first signal
component was set to a measurement noise of 60 m of standard deviation, which
increased for the following components by 30 m/µs.
Positioning results with outdoor and indoor signals
The receiver’s positions were calculated with indoor and outdoor measurements, for
every possible combinations of NLOS mitigation algorithms and TOA estimators
presented in the previous sections. The results are presented in Universal Transverse
Mercator (UTM) coordinates projected on the WGS-84 ellipsoid. The sources of
CDMA2000 measurements were the same as the ones used for the stability experiments
presented in Section 5.2.
−7500 −5000 −2500 0 2500 5000
Easting (m)
−7500
−5000
−2500
0
2500
5000
N
or
th
in
g
(m
)
80
164
180
232
356
440
Figure 5.6: Map of the base stations (blue triangles) and the receiver (red cross)
positions. The coordinates are in the UTM coordinate system and centred on the
position of the receiver CDMA2000 antenna.
The transmitter geometry is represented in Figure 5.6. The vertical position of
the antennas at the BTS was uncertain. Some antennas were located at the top of
pre-existing buildings and others were attached on masts anchored at ground level.
I used Google Maps (Google Maps 2012) to take the altitude of the base stations. The
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5. GNSS aiding in synchronised cellular networks
heights of the centres of phase of the antennas, above ground level, were estimated
from photographs. We can assume this leads to a systematic vertical error of about
100 m.
The Cramér-Rao bound of the position estimator is calculated from Equation 2.19
in Chapter 2, under the assumption that the signals from all seven BTSs have been
detected. The noise variances of the pseudo-range measurements from the different
BTSs are taken from the lower bounds calculated in Table 5.3. Fewer signals are detected
when the antenna is indoors and the CRLB of the position estimate becomes wider.
Table 5.4: Statistics of the position errors using outdoor CDMA2000 signals. The
circular error probable is calculated after the median bias has been removed from
the position error; the CEP is relative to the median position estimated and not to
the true position.
Multipath TOA Filter Median bias CEP50% CEP90%
mitigation estimator (m) relative (m) relative (m)
Residuals Single-path No 1 23 49
Residuals Single-path Median 1 15 32
Residuals Heuristic No 1 23 49
Residuals Heuristic Median 0 15 32
Residuals SAGE No 1 23 50
Residuals SAGE Median 0 15 32
Full enumeration Single-path No 3 46 164
Full enumeration Single-path Median 1 26 65
Full enumeration Heuristic No 4 46 173
Full enumeration Heuristic Median 2 27 67
Full enumeration SAGE No 1 46 172
Full enumeration SAGE Median 4 27 73
The position errors using outdoor signals are summarised in Table 5.4. I used the
circular error probable (CEP) to analyse the position errors. Different definitions of
the CEP may exist and I defined the CEP as the radius of the circle that included a
given percentile of errors, after removal of the bias. The CEP50% was for the median
of the errors and the CEP90% for the 90-percentile of the errors. I did not observe any
significant bias. The circular error was between 20 m and 50 m and the median filter
reduced this error to between 15 m and 30 m.
I then re-used the set of indoor measurements described in Section 5.2 to analyse the
performance within an indoor environment. The true locations of these measurements
were not known with an accuracy better than 10 m. I considered that the true position
of the indoor antenna was the same as the one of the outdoor CDMA antenna, minus
6 m in height to reflect the location of the floor where the experiments were conducted.
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5.5. Positioning using cellular signals only
The position errors presented in this section may thus have an error component caused
by the uncertainty in the true position of the measurements, which is bounded by the
largest dimension of building, about 20 m. The analysis still holds as the uncertainties
in the true positions were dominated by the larger errors in the TOA estimations.
Table 5.5: Statistics of the position errors using indoor CDMA2000 signals. The
circular error probable is calculated after the median bias has been removed from
the position error; the CEP is relative to the median position estimated and not to
the true position.
Multipath TOA Filter Median bias CEP50% CEP90%
mitigation estimator (m) relative (m) relative (m)
Residuals Single-path No 347 278 577
Residuals Single-path Median 417 262 506
Residuals Heuristic No 481 328 596
Residuals Heuristic Median 536 301 553
Residuals SAGE No 61 153 635
Residuals SAGE Median 33 121 589
Full enumeration Single-path No 20 222 1703
Full enumeration Single-path Median 74 185 693
Full enumeration Heuristic No 140 283 1043
Full enumeration Heuristic Median 167 223 827
Full enumeration SAGE No 78 249  10 km
Full enumeration SAGE Median 26 177 805
−2 0 2 4 6 8 10 120
100
200
300
400
500
600
(a) Position error bias (m).
−2 0 2 4 6 8 10 120
50
100
150
200
250
300
350
(b) CEP50% (m).
Residuals rejection, Maximum
Residuals rejection, All peaks
Residuals rejection, SAGE
Full enumeration, Maximum
Full enumeration, All peaks
Full enumeration, SAGE
(c) Legend. The hatched bars rep-
resent the performance after me-
dian filtering.
Figure 5.7: Summary of median position errors for all combinations of detectors
and positioning methods.
The position errors using the indoor signals are summarized in Table 5.5 and
Figure 5.7. The smallest bias was obtained for the full enumeration of all the peaks
detected in the likelihood ratio. This combinatorial method was the most computation-
ally expensive and it did not provide the best median error. A combination of SAGE
estimation and residual rejection was the second-best method in terms of bias and it
offered the smallest relative position errors. The median filter increased the bias error
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5. GNSS aiding in synchronised cellular networks
−400 −200 0 200 400
UTM easting error (m)
−300
−200
−100
0
100
200
300
U
TM
no
rt
hi
ng
er
ro
r
(m
)
(a) Residual rejection with SAGE detector.
−400 −200 0 200 400 600 800
UTM easting error (m)
−200
−100
0
100
200
300
400
U
TM
no
rt
hi
ng
er
ro
r
(m
)
(b) Residual rejection with maximum detector.
Figure 5.8: Position errors of indoor signals, presented as 2D-histograms, with
rejection of measurements associated to large residuals. The colour of each cell
indicates the frequency of the position solutions calculated inside the cell.
of all positions except when the SAGE estimator was used. A possible reason is that the
position estimate was systematically biased because of the NLOS signals. The median
operation reduced the estimator variance, as shown by the smaller median error, but
did not improve the estimator bias.
Figures 5.8 and 5.9 show that the shapes of theoretical CRLB curves are in good
agreement with the observed errors. We also note that the positions errors with the full
164
5.5. Positioning using cellular signals only
−400 −200 0 200 400 600 800
UTM easting error (m)
−300
−200
−100
0
100
200
300
400
500
600
U
TM
no
rt
hi
ng
er
ro
r
(m
)
(a) Full enumeration with maximum detector.
−1000 −500 0 500
UTM easting error (m)
−600
−400
−200
0
200
400
600
U
TM
no
rt
hi
ng
er
ro
r
(m
)
(b) Full enumeration with SAGE detection.
Figure 5.9: Position errors of indoor signals, presented as 2D-histograms, using the
full enumeration of measurement combinations. The colour of each cell indicates
the frequency of the position solutions calculated inside the cell.
enumeration methods concentrated in a few discrete locations, whereas the residual
rejection method had a more even error distribution.
Conclusions
The SAGE-based estimator combined with a residual rejection method provided the
best position accuracy and computational efficiency for my series of measurements. The
165
5. GNSS aiding in synchronised cellular networks
position error at 50% was about 20 m with no bias using outdoor signals. As expected,
indoor positioning with CDMA2000 signals was more challenging. A systematic bias
of about 60 m was observed and the position accuracy, corrected for the bias, was about
150 m at the 50-percentile. It is unlikely that the systematic bias, introduced by the
NLOS signals, can be corrected in a snapshot scenario, even with more sophisticated
algorithms. Tracking of the measurements over a longer period of time is required to
gather more information about which signals are LOS or NLOS.
5.6 Positioning using GNSS and cellular signals
This section presents the evaluation of a positioning algorithm that combines GPS and
CDMA2000 signals.
Challenges of GNSS and cellular signals combining
Combining the times of arrival of the CDMA2000 and GPS measurements is straight-
forward since the two systems are synchronised together. The GPS measurements can
provide additional pseudo-range measurements that can be included in the position
equation. This approach has several advantages.
We have seen in the previous section that the detection of CDMA2000 signals can
be challenging indoors and three measurements are not always available. The added
signals improve the least-squares solution, both in terms of accuracy and availability.
Furthermore, signals in an indoor environment with one entry point, such as a
window, can have a large range of signal strengths. In CDMA systems, this can
introduce the near-far problem, since the noise floor at the receiver input is increased
by the spread-spectrum modulation of the stronger signals, which mask the more
attenuated signals. I did not observe the near-far problem in these experiments since the
receiver had a large dynamic range. However, this can be problematic in mass-market
mobile receivers and it can reduce the number of base stations available for positioning.
Another advantage of using GPS measurements is that it adds diversity to the
propagation conditions of the signals. GPS signals typically arrive from high elevations,
whereas CDMA2000 ones generally propagate in a horizontal plane. The GPS signals
thus add some vertical components to the problem geometry. The GPS signals may
also be less affected by multipath propagation, although this depends on their angles
of arrivals. The GPS L1 C/A signal has a bandwidth of about 20 MHz, compared with
1.2288 MHz for CDMA2000. Wider bandwidths improve the TOA estimation and may
be beneficial to mitigate multipath errors. However, most mass-market GPS receivers
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5.6. Positioning using GNSS and cellular signals
have a narrower bandwidth, generally around 2 MHz, and the L1 C/A signal is not
designed for precise positioning. It is therefore unlikely that the wider bandwidth of
GPS signals be a source of a performance improvement.
A drawback of assisted-GNSS receivers with millisecond-ambiguity positioning,
described in Section 2.5, is the need to estimate the signal transmission times, which
adds another unknown variable to the position equation. This may be a problem for the
availability of positioning in indoor environments, where enough signals may not be
detected. However, CDMA2000 networks are synchronised with GPS time, hence the
mobile device knows an approximation of the GPS system time. The only uncertainty
in the GPS system time is related to the unknown propagation time of the CDMA2000
signal carrying the GPS time. I will show that the time information provided by the
CDMA2000 network is accurate enough to calculate the times of transmission of the
GPS signals, thus removing one unknown variable from the position equation.
The method presented in this section might be similar to the Hybrid-AFLT system
commercialised under the name gpsOne™ by Qualcomm Inc. Biacs et al. (2002) give some
general information about this system, however most of the details seem proprietary.
Other research on the fusion of cellular and GNSS signals includes Mensing et al. (2010),
who present simulation results of positioning using a combination of GPS and cellular
4G LTE signals. Their method includes signal tracking to improve the position solution
over time and does not address NLOS mitigation.
Extraction of GPS time from CDMA2000 data
The CDMA2000 BTSs transmit the times of their clocks in the Synch Channel Messages
(SCHM), as described in 3GPP2 (2004b, §3.7.2.3.2.26). The time resolution of this
message is 80 ms and its transmission is aligned with the transmission of the pilot code.
The SCHM also carries information about the base station identification, the PN offset
index, and the carrier frequency. I describe my implementation of the SCHM decoder
in Appendix B.
Positioning with pseudo-range combining
I now derive the position equation combining the CDMA2000 and GPS signals. The full
GPS pseudo-ranges can not be calculated since the GPS signals are too weak to allow
decoding of the navigation data. Only fractional measurements, with an ambiguity of
the period of the L1 C/A code, TCA = 1 ms, are available. The solution to this problem
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5. GNSS aiding in synchronised cellular networks
is given by the millisecond-ambiguity method introduced in Section 2.5, which seeks to
resolve the full pseudo-range of the signals and determine the satellite positions.
The first step is to resolve the millisecond ambiguity in the L1 C/A signals, which
requires knowing the GPS system time with an accuracy better than 500µs at the
receiver location. The maximum range of BTS is about 15 km, about 50µs, and we have
seen the CDMA2000 signals are synchronised with the GPS time with an accuracy of
about 2µs. The GPS time, from the CDMA2000 Sync Channel Message, is thus known
at the receiver location with enough accuracy to resolve the millisecond ambiguity.
The second step is to calculate the time of transmission (TOT) of the L1 C/A signal
in order to determine the satellite positions. The time of transmission is calculated from
the time-of-week word of the navigation data in a receiver with tracking loops. This
information is not available to our high-sensitivity receiver and the time of transmission
must be estimated by the receiver. This requires an additional unknown variable to
estimate this time. However, the CDMA2000 Sync Channel Message can provide the
GPS time with an uncertainty which is caused only by the unknown propagation
time and is bounded by a maximum error of 50µs. The maximum range rate of
the GPS is about 465 m/s, therefore the pseudo-range error caused by the uncertain
satellite position is less than 0.023 m, which is perfectly acceptable in our application.
This eliminates the need for an additional unknown variable, thus making more
measurements available to solve the position equation. The satellite ephemeris is
downloaded from the International GNSS Service (see Dow et al., 2009).
The position equation needs to be altered since the signals, although synchronised,
have spreading codes with different periods. We consider a set of i = 1, . . . ,Ntx
transmitters, sending positioning signals with possibly different periods. The time
relationships between the code periods and the propagation times are represented
schematically in Figure 5.10, with: TOTi, the time of transmission of the signal from
transmitter i; TORi, the time of reception of the signal; t0, the time origin of the receiver
time scale; and talign, the last time at which all the positioning codes were aligned. The
goal is to calculate the true, unknown, range ||xi − x||. We neglect the clock bias of the
transmitters, which can be simply applied as a correction to TOR. From the diagram,
we can deduce the following relationships:
||xi − x|| =TORi − TOTi
= (TORi − t0)︸       ︷︷       ︸
code phase
+ (t0 − tˆ0)︸  ︷︷  ︸
local time error
+ (tˆ0 − talign)︸      ︷︷      ︸
new local time origin
− (TOTi − talign)︸           ︷︷           ︸
code period ambiguity
= τˆi − bu + tˆ0 mod Tlcm − ki Ti
(5.2)
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5.6. Positioning using GNSS and cellular signals
τˆCA
Tpilot
τˆPN
t0
TCA
||xCA − x||
tˆ0
TOTCA TORCA
TOTpilot TORpilot
bu
∣∣∣∣∣∣xpilot − x∣∣∣∣∣∣
Figure 5.10: Timing diagram of CDMA2000 and GPS synchronisation signals. The
unknown variables are represented in green and the measurements in blue. The
subscripts pilot and CA are used to label the variables related to the CDMA2000
and GPS systems, respectively.
with
• Ti, the code period of the signal from transmitter i;
• τˆ, the measured code phase;
• bu, the unknown receiver time bias;
• Tlcm, the shortest period that all the ranging codes have in common;
• tˆ0, an estimate of the GPS system time at the beginning of the measurements; and
• ki, the unknown integer number of code periods since the latest instant when the
signals were aligned.
The position equation, Equation 5.2, is similar to the position equation with
millisecond-ambiguity positioning presented in Chapter 2, except that the receiver
bias bu is substituted by the corrected bias (bu − tˆ0) mod Tlcm. Formally, Tlcm =
LCM
(
TCA,Tpilot
)
where LCM(x, y) is the least common multiple of x and y. For
the GPS L1 C/A and CDMA2000 PN pilot signals, we have Tlcm = 80 ms. tˆ0 is required
with an accuracy better than the smallest signal code period in order to resolve ki
unambiguously. We showed this was possible in the previous section, using the
CDMA2000 system time given in the Sync Channel Message.
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5. GNSS aiding in synchronised cellular networks
Indoor GPS signal acquisition
Detecting the synchronisation parameters of indoor signals requires long signal in-
tegration periods. Agarwal et al. (2002) highlight the importance of maximising the
length of the coherent integration period in order to maximise the probability of
detection, although this increases the search space size and the acquisition time, as
seen in Chapter 3. We can, however, effectively reduce the search space since we have
fine-time aiding from the CDMA2000 network. The larger search space caused by the
finer resolution is thus balanced by a smaller total size, hence the time-to-first-fix can
remain the same.
The bit flips of the navigation message limit the coherent integration time to a maxi-
mum of 20 ms for the GPS L1 C/A signal. Advanced high-sensitivity techniques, such
as the ones described by Ziedan (2006), can extend the coherent integration time beyond
the length of one navigation bit, although this generally requires computationally-
demanding algorithms. Fine-time aiding makes these methods more tractable. The full
pseudorange is available with fine-time aiding, hence the bit edges can be estimated
within a few hundreds of microseconds, depending on the initial uncertainties about
the receiver time and position. It then becomes possible to correct the phase shifts
introduced by the navigation message and extend the coherent integration time, a
technique often called “data wipe-off”. I did not perform any data wipe-off because
some of the experiments were carried out while walking. The changing Doppler
frequencies caused by the human motion would have decreased the sensitivity of the
receiver using long coherent integration times.
I chose a coherent integration time of 3 ms, which may not be optimal but provides
reasonable performance for this evaluation. The result of the coherent integration was
squared and 330 further non-coherent integrations were performed, resulting in a total
integration time of 990 ms. This integration time provided a near-certain probability of
detection for signal strengths down to 24 dB−Hz, using a threshold set for a false-alarm
rate of 10−4. I noticed the noise floor at the input of the GPS indoor channel varied with
the location of the indoor antenna. I disabled the automatic gain control of the signal
analyser and used an adaptive threshold in order to keep a constant false-alarm rate. I
chose an FFT-based acquisition of the GPS signals, as explained in Chapter 2, which
used the full code-phase search space. The range of possible codes phases was 0 to 1 ms
and the resolution was given by the sample rate. The frequency offset was calculated
from the approximate receiver position, the known bias of the local oscillator, and the
approximate GPS system time provided by the CDMA2000 network. The search space
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5.6. Positioning using GNSS and cellular signals
was thus composed of one frequency offset and 10 000 possible code phases.
Results of indoor positioning
The time biases of the times of arrival of the PN pilot signals were calibrated in Section 5.3
to ensure their synchronisation with GPS time. This calibration was performed using
the SAGE estimator and I only employed this estimator in the rest of this section.
Combining GPS and CDMA2000 signals required perfect synchronisation of the time
scales of both systems, hence I applied the same estimator for both the calibration
and the experiment. I only investigated the combination of indoor signals since the
performance using only CDMA2000 outdoor signals was already satisfactory.
−100 −50 0 50 100 150
UTM easting error (m)
−200
−150
−100
−50
0
50
100
150
U
TM
no
rt
hi
ng
er
ro
r
(m
)
Figure 5.11: 2D-histogram of the position errors with indoor signals, CDMA2000
and GPS combining, with SAGE detector and residuals rejection.
Table 5.6: Error statistics of positioning with CDMA2000 and GPS signals, indoors.
Multipath TOA Filter Median bias CEP50% CEP90%
mitigation estimator (m) relative (m) relative (m)
Residuals SAGE No 11 64 282
Residuals SAGE Median 12 54 204
The histogram of the position errors is presented in Figure 5.11 and the error statistics
are summarised in Table 5.6. The performance of positioning using a combination of
CDMA2000 and GNSS signals was approximately three times better than positioning
using CDMA2000 signals only: the bias was reduced from about 30 m to about 10 m,
and the CEP50% was reduced from about 150 m to about 50 m.
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5. GNSS aiding in synchronised cellular networks
5.7 Discussion
Indoor positioning improvement with signal combining
0 2 4 6 8 10
Number of measurements
0
200
400
600
800
1000
1200
1400
Fr
eq
ue
nc
y
(a) Histogram of the number of measurements ef-
fectively used by the position algorithm. The blue
bars represent the CDMA2000 measurements only
and the green bars the combination of GPS and
CDMA2000 signals.
3 4 5 6 7 8 9
Nb. of measurements
50
100
150
200
250
300
350
400
M
ed
ia
n
er
ro
r
CDMA2000 only
CDMA2000 and GPS
(b) Effect of the number of measurements on the
position error. The green curve includes all types
of measurements, it does not attempt to match the
CDMA2000 measurements with those used for the
blue curve.
Figure 5.12: Comparison of the availability and accuracy between CDMA2000-only
positioning and CDMA2000 and GPS combining.
Combining CDMA2000 and GPS signals improved the accuracy of the positioning
when indoors, as well as the availability. Tables 5.5 and 5.6 show that the position
error had a systematic bias of 61 m and a median relative error of 153 m with only
CDMA2000 measurements, whereas the bias reduced to 11 m and the median error to
64 m with both CDMA2000 and GPS signals when indoors.
Figure 5.12 shows the relationships between the number of measurements used
to calculate a position and the position error. The number of detected signals was
generally higher than the number of signals presented in Figure 5.12, but some of
the detected measurements were rejected by the NLOS positioning algorithms if they
introduced large residuals. The median error of CDMA2000-based positioning was
almost independent of the number of measurements used. This may have been related
to the presence of a strong multipath component on BTS 440, which was included in
almost all calculations. Although the position accuracy with three measurements did
not seem to be improved by adding GPS signals, the three measurements were more
likely to come from GPS signals, which had a poor geometry. The probabilities of
detection of some of the GPS signals were higher than those of the CDMA2000, possibly
because the signals from the satellites were less affected by multipath fading than the
signals from the BTSs.
The set of indoor measurements was composed of sequences taken on two different
172
5.7. Discussion
Sta
tic
1
Sta
tic
1
Sta
tic
1
M
ov
in
g
M
ov
in
g
M
ov
in
g
Ha
nd
-h
eld
Sta
tic
2
Time, not continuous (s)
PRN 1
PRN 32
BTS 80
BTS 440
Figure 5.13: Signal detection indoors. The different sequences of measurements
are juxtaposed in the diagram, hence the time scale is not continuous. The “Static1”
and “Moving” sequences were performed on a same day, whereas the “Hand-help”
and “Static2” experiments were carried out on a different day.
days, as seen in Section 5.2. The total duration of the experiment during each day
was about 10 minutes, which was short enough to neglect the changes in the GPS
constellation geometry. However, it was difficult to summarise the results for both
days, since the geometry changed between these days. I analysed the geometry of
the transmitters for the first day of measurement only. The high-sensitivity receiver
generally detected 4 to 5 satellites successfully when the receiver was static. An
exhaustive analysis of the geometry of the transmitters and its effects on the DOP were
difficult since we used an algorithm that could reject some transmitters at every time
step. The varying sets of transmitters meant the DOP was not constant. Figure 5.13
shows the evolution of the measurements used by the positioning algorithm for the
different series of experiments. We note that the detection and use of the signals from
the BTSs were generally constant. On the other hand, the acquisition of GPS signals was
more difficult when the antenna is moving. The coherent integration time of 3 ms was
not a limiting factor as the antenna was moved slowly and the Doppler shift remained
small. It is more likely that the multipath fading reduced the GPS signal strength too
much to ensure a reliable acquisition at every time step.
The horizontal dilution of precision (HDOP) is frequently used to relate the root mean
squared of the position errors and the geometry of the transmitters. The relationship
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5. GNSS aiding in synchronised cellular networks
between the root mean squared and the CEP50% is linear and, for a bivariate normal, we
have the RMS = 1.2CEP50% see Diggelen, 2007. The HDOP can therefore be also used
to compare the 50% CEP. However, the dilution of precision assumes a Gaussian noise
on the pseudo-range measurement, which is not the case in our experiments because
of the NLOS signals. Besides, the sample root mean squared of the position errors is
affected by a few outliers and can not be computed accurately.
The comparison between HDOPs may therefore be unreliable. However, it may help
coarsely quantify the contribution of the geometry to the improvement in the position
error we observed between the two experiments. The HDOP for the CDMA2000
transmitter geometry was 1.89 and it was 1.06 for the CDMA2000 and GPS transmitters
geometry, taking the four GPS satellites that were the most frequently detected. The
inclusion of the GPS signals should therefore have reduced the standard deviations of
the position errors by about 43%, simply by improving the number of transmitters and
their geometry. Instead, we observed a 60% improvement in the CEP50%, thus showing
that the addition of GPS satellites had more benefits than we could expect from the
theoretical performance.
Non-line-of-sight mitigation in snapshot receivers
We have only addressed the case of snapshot receivers, for which only 500 ms worth of
measurements is available. Figure 5.5 shows that the durations of the fades, also called
the coherence time of the channel, was typically of a few seconds for the measurements
made on a static receiver indoors. The duration of the measurements used by the
snapshot receiver was thus not long enough to identify and filter the non-line-of-sight
components.
Multipath mitigation algorithms that exploit measurements taken over durations
longer than the coherence time of the channel are examples of time diversity techniques.
An alternative to time diversity is space diversity, which exploits the differences in
the propagation characteristics of a channel at different locations. Space diversity is
commonly used in radio communication systems to mitigate the effects of multipath
fading. Dietrich et al. (2001) show that combining the signals received from two
antennas often provide an average gain of 10 dB above the signal at one antenna only.
Space diversity thus mitigates the effect of multipath fading, which can improve the
detection of indoor signals in a fading channel. I have not had a the opportunity to try
space diversity techniques, however I expect that the detection rate of the signals when
walking with the antenna, shown in Figure 5.13, would be significantly higher. The
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5.8. Conclusions
higher detection rate would give more pseudo-range measurements to the positioning
algorithm and thus improve the position accuracy.
Most multi-antenna receivers used in positioning systems perform more advanced
processing than simply combining the signals. Antenna arrays have been used in GNSS
receivers to measure the angles of arrival of the signals, which can be used to identify
jamming signals for example. More importantly, determining the angles of arrival of
the signals can help identify the non-line-of-sight components. Broumandan et al. (2008)
have performed measurements on a CDMA2000 network and used the MUSIC super-
resolution algorithm to identify the NLOS signals. Their results suggest that the antenna
array is particularly beneficial in situations were few pseudo-range measurements,
typically three, are available. However, the space required for an antenna array with
more than two antennas is often not compatible with hand-held devices, as the antennas
must be separated by about the wavelength of the radio-frequency carrier.
Positioning with accurate clocks
Further research could exploit the accuracy of clocks to improve the signal acquisition
performance, a remark that was also made in Chapter 4. The time accuracy of the
CDMA2000 signals enables the receiver to know the bit edges of the navigation message.
This opens the way to using coherent integration times that are longer than the code
period of the PN signal, which improves the sensitivity of the receiver. I also mentioned
that direct-position estimation techniques, as proposed by Closas et al. (2007) and
Axelrad et al. (2011), must deal with the addition of the local oscillator bias to the search
space. Accurate time thus reduces the search space and facilitates the computation of
the position directly from the received signals.
5.8 Conclusions
In summary, it is difficult to address the performance degradation caused by non-line-
of-sight signals in a snapshot receiver. The most common mitigation methods assume
that tracking the signals is possible and the accumulation of measurements help identify
and mitigate the NLOS conditions. A snapshot receiver only has a few milliseconds
worth of measurements, which is often much shorter than the coherence time of a
multipath channel, hence the NLOS conditions can not be identified. A more promising
approach, for snapshot receivers, is to increase the number of signals employed by
the positioning algorithm. I showed two methods in this chapter that can increase the
number of signals available. The first method consists of combining measurements
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5. GNSS aiding in synchronised cellular networks
from GNSS and cellular systems. The second method consists of using an accurate
estimate of the GPS system time provided by the cellular network to remove one of
the unknown variable in the position equation, thus increasing the availability of the
solution to fewer measurements.
176
CHAPTER 6
Conclusions
6.1 Summary of contributions
The work presented in this thesis focused on the use of GNSS receiver in cellular
networks. As such, we have only considered a “push-to-fix” scenario, explained in
Chapter 2, a situation which had not been addressed frequently in the literature. The
main contributions of this work are both theoretical and experimental.
We have first seen, in Chapter 3, a new mathematical framework to calculate a
lower bound of the time-to-first-fix of assisted-GNSS receivers. This work was based
on the flow-graph method already proposed in the literature, but which was limited
to the mean acquisition time of a single satellite. The framework consists in a flexible
collection of several mathematical tools, such as the probability generating function,
that can be reused to model the acquisition process of different types of GNSS receivers.
I applied these tools to the case of an assisted-GPS receiver and I studied the impact
of a fading channel on the acquisition time. One of the results is that the TTFF can be
significantly improved by the presence of a single strong signal, a phenomenon which
is not observed in non-assisted receivers. The lower bound also provides some insight
about the expected performance of a receiver, which can be useful during the testing
and verification of receivers.
Chapter 4 contained the results of experiments about the time stabilities of the
synchronisation signals in GSM networks. The results showed a large heterogeneity
in the stabilities of the signals from different base stations. Some base stations had a
stability better than a few microseconds over several days, while other had variations of
hundreds of milliseconds within one day. However, the measurements showed that the
177
6. Conclusions
base stations were synchronised neither between themselves, nor with an external time
reference. They were therefore not capable of providing universal time information,
which is required in order to provide time assistance to a GNSS receiver. I developed
an ensemble filter which maintains a single estimate of the universal time by exploiting
the time stability of the GSM synchronisation signals. The absolute time of the filter
was periodically updated by calculating the GPS position and time. The results, using
a set of twelve base stations in Cambridge, UK, showed that such a filter can provide
fine-time aiding with an accuracy better than 10µs.
The case of synchronised network was addressed in Chapter 5, where I report the
results of experimental on a CDMA2000 network. The CDMA2000 standard specifies
that the network clocks must be synchronised within 2µs of the GPS time, but the
standard does not specify the stability. The measurements showed that the CDMA2000
network I considered was both stable and accurately synchronised with the GPS time.
The synchronisation and the stability were then exploited to derive pseudo-range
measurements from the times-of-arrival of the CDMA2000 pilot signals. The user’s
position was calculated with a median circular error of about 20 m in an outdoor
environment. The non-line-of-sight measurements degraded the performance in an
indoor environment, even after applying a mitigation algorithm, and the indoor position
had a bias of about 60 m and a median relative error of 150 m. The pseudo-range
measurements from the CDMA2000 base stations were combined with the fractional
pseudo-ranges from the GPS satellites, which were typically used in mobile devices
that require a short time-to-first-fix. I detailed how the position equation can be solved
when using pseudo-noise signal of different code periods. Both the accuracy and the
availability were significantly improved by the addition of GPS measurements.
6.2 Future work
Different points of the methods developed in this thesis could be improved in future
works. For example, a more accurate model of false alarms in flow-graph methods
could be investigated. The results presented in Chapter 3 could be compared with the
performance of simpler filter, median or moving average filters for instance. All these
improvements require more detailed assumptions about the receiver structure, cellular
network architectures, user expected performance and trade-offs, etc. that are beyond
the scope of this thesis, but that could be discussed in a follow-up work focused on the
practical implementation details.
Third and fourth-generation cellular networks, for example W-CDMA and LTE,
178
6.2. Future work
have larger signal bandwidths, thus offering more possibilities of multipath mitigation.
A promising research area is the joint positioning approach proposed by Closas
et al. (2007) and Axelrad et al. (2011). The most common acquisition methods of GNSS
signals, which we have seen in this thesis, has two steps: first, the maximum-likelihood
estimates of the synchronisation parameters are calculated using a GLRT, then the
position is calculated from the synchronisation parameter estimates. On the other
hand, joint positioning is a one-step method, where the likelihood function of the
user’s position is calculated directly from the observed signals. The signals are thus
combined together and even weak signals, below the detection threshold of a classical
GLRT detector can contribute to the position solution. This approach has not been
used commercially yet, possibly because it is more computationally demanding and
because classical methods work well in good conditions. However, I think that GNSS
receivers could benefit from joint positioning, in particular in challenging situations
such as operating with weak signals in multipath conditions. One of the problems with
joint positioning is that the search space of the unknown parameters, the three spatial
coordinates and the user bias, is larger than the search space of the synchronisation
parameters used in the classical approach. The work on fine-time aiding presented in
this thesis could help reduce the search space by providing an accurate estimate of the
user clock bias. The search space of the joint positioning solution can even become
smaller than the search space of a multi-constellation receiver. Joint positioning may
also resolve some problems caused by multipath propagation, since it requires the
full likelihood function of the pseudo-range, not simply the most likely value of the
pseudo-range. The multipath components that appear in the likelihood function of one
signal can therefore by partially reduced by the likelihood values of another signal.
179

APPENDIX A
GPS software receiver
The GPS software receiver designed and used throughout this thesis is presented in
this section. The software receivers used to receive the GSM and CDMA2000 signals
were very similar to the one presented here.
A.1 Acquisition
A schematic representation of the acquisition stage of the GPS receiver is shown in
Figure A.1.
The input signal was sampled at baseband with a period Ts and a block of Ns
complex samples was acquired. The samples acquired are represented by the vector z.
The signal was first down-converted to remove the frequency offset, f , caused by the
Doppler effect and the local oscillator bias.
The coherent integration period was Tcoh and it was a multiple of the code period,
TCA, of the GPS L1 C/A signal, such that Tcoh = NcohTCA. The sampling period, Ts was
Ncoh∑
z
FFT∗
FFT FFT−1
Knc∑ |.|2 SKnc( f , τ)
f
z
Figure A.1: Block diagram of the software GPS receiver.
181
A. GPS software receiver
chosen such that there was an integer number of samples, NCA, in a code period. The
sampling period was also chosen so that NCA was a power of two (or at least close to a
power of two), in order to use the optimised fast Fourier transform implementation of
the DFT.
The linearity property of the discrete Fourier transform allowed us to perform the
coherent integration before removing the code phase. The length of the DFT was thus
kept to NCA samples, which made it easier to optimise the implementation of the FFT.
Signal vectors of length NCA were added together Ncoh times to form the vector of
integrated signal zcoh. The receiver architecture was optimised by setting the number
of samples acquired, Ns, such Ns = NcohNCA.
The acquisition was performed following the fast Fourier transform (FFT) method
proposed by Davenport (1991). This method exploits the dual relationship between
the circular correlation and the product operation when using the discrete Fourier
transform. The advantage was that the FFT, with a computational complexity of
O(N log2 N), was more computationally efficient than the correlation operation, which
has a complexity of O(N2). The FFT was implemented using the FFTW library.
A vector composed of the NCA samples of discrete Fourier transform of the C/A code,
zCA, was stored for the duration of the acquisition. The conjugate of the correlation
vector was then multiplied with the results of the FFT of the input vector after coherent
integration. The value of the log-likelihood ratio was calculated by squaring the
magnitude of the vector of correlated samples. The result of this operation formed
the output values of S0(τ, f ), which was calculated for a fixed f and for all the discrete
code phases calculated by the FFT. The acquisition process was repeated for another
frequency offset candidate, f , at a distance ∆ f of the previous frequency offset.
The correlator output, S1(τ, f ), was thus calculated with a resolution of ∆ f along
the frequency-offset axis, and with a resolution of Ts along the code-phase axis. It is
worth pointing out that the FFT approach calculates the correlation values for all the
code-phase values in the discrete search space. The FFT does not scale well when only
the values for a subset of the code phase search space are required.
A.2 Tracking
The frequency offset estimate, fˆ , was tracked using a phase-locked loop (PLL) and the
code-phase estimate, τˆ, was tracked using a delay-locked loop (DLL). The frequency
offset was first refined using a frequency estimator similar to the one described by Borio
et al. (2008a). This estimator also provided the phase offset of the signal. The phase
182
A.2. Tracking
offset and the improved frequency estimate helped the convergence of the phase-locked
loop. The code-phase estimate was already accurate enough for the good operation of
the DLL.
The tracking loops were implemented using the C programming language and were
optimised using look-up tables of the trigonometric functions and vectorised functions
from the Basic Linear Algebra Subprograms (BLAS) library. The frequency estimate
from the acquisition stage was refined by using a frequency estimator, which helped the
convergence of the PLL. We only require off-line, not real-time, operation of the receiver
hence the synchronisation parameters of the different signals were tracked sequentially.
The values of the code phase, frequency offset, and carrier phase estimated by the
tracking loops were then stored on disk for further processing.
183

APPENDIX B
CDMA2000 Sync Channel Message
The CDMA2000 standard is complex and understanding the reference documents can
be time consuming. This appendix contains the information relevant to the decoding
of the Sync Channel Message, from the encoding of the physical layer to the structure
of the information in the bitstream.
B.1 Physical channel
The physical layer is described in the document 3GPP2 C.S0002. The logical channel
F-SYNC is formed by successively applying the following operations:
1. convolutional encoding of rate k = 2, with the polynomials 753 (octal) and 561
(octal);
2. symbol repetition of rate 2;
3. block interleaving of length 128 symbols, based on a bit reversed-order permuta-
tion table;
4. symbol mapping at a rate of 4800 bit/s and spreading by the Walsh sequence 32,
at a rate of 1.2888 Mchips/s;
5. spreading by the IS95 pilot pseudo-noise code, shifted by the index offset of the
base station; and
6. modulation using binary phased-shift keying and pulse shaping.
The demodulation is performed by correlation with the conjugate of the PN sequence,
despreading by the Walsh sequence, coherent integration to recover the 128 symbols
at a rate of 4800 bit/s, de-interleaving using the permutation table, further coherent
integration to remove the symbol repetition, and finally convolutional decoding using
the Viterbi algorithm.
185
B. CDMA2000 Sync Channel Message
The Walsh sequence 32 is equal to the 32nd row of the Hadamard matrix of order
64. In this case, the Walsh sequence, represented as a non-return-to-zero (NRZ) signal,
is simply a succession of 16 bits +1, followed by 16 bits −1, i.e. {+1, . . . ,+1,−1 . . . ,−1}.
This pattern is a common property of the n/2 row of any n-order Hadamard matrix.
B.2 Frames and super-frames
The next step is to recover the F-SYNC channel frames, as described in the 3GPP2
C.S0003 and C.S0004 documents. One frame is composed of 32 bits, the first bit being
the Start Of Message bit (SOM) and the remaining 31 bits are the data bits.
Three frames form of super-frame, and an integer number of super-frames form a
message capsule. A message capsule always starts by a frame where the SOM bit is
set to one. All the SOM bits are removed from the frames and only the data bits are
assembled to form the data bits of the capsule.
The capsule carries data packets with the following format, described in C.S0004:
• bits 0 to 7 contain the Packed Data Unit (PDU) length (MSG_LENGTH), in bytes;
• bits 8 to 15 contain the message type, composed of 2 bits for the Protocol Descriptor
(PD) and 6 bits for the message ID (MSG_ID).
• bits 16 until MSG_LENGTH bytes contain the PDU;
• at the end of the PDF, 30 bits are used for by the Cyclic Redundancy Check (CRC)
value; and
• after the CRC bits, a variable number of bits with value 0 are used to pad the
capsule to the size an integer number super-frames.
On the F-SYNC channel, the Sync Channel Message is assigned the MSG_ID tag
SCHM, of value binary value 000001.
B.3 Sync Channel Message structure
Finally, the content of the Sync Channel Message is described in 3GPP2 C.S0005 (section
3.7.2.3.2.26 of revision D). Of interest are:
• the NID and SID fields contain the network and the base station identifiers;
• the PILOT_PN field contains the base station offset of the PN code;
• the SYS_TIME field contains the system time (defined to be GPS time), with a
resolution of 80 ms;
• some other fields contain the number of leap seconds, time zone offset and
daylight saving time;
186
B.3. Sync Channel Message structure
• the CDMA_FREQ field contain the frequency of the base station, using the CDMA
frequency code described in 3GPP2 C.S0057;
• other fields contain protocol revision information, spreading rate information,
etc.
187

References
3GPP. 3GPP - LTE. 3GPP LTE website. url: http://www.3gpp.org/LTE (visited on
28/11/2012) (cit. on pp. 30, 136).
— (July 2003a). Multiplexing and Multiple Access on the Radio Path. TS 05.02. 3rd
Generation Partnership Project (3GPP) (cit. on p. 98).
— (Sept. 2003b). Radio subsystem synchronization. TS 05.10. 3rd Generation Partnership
Project (3GPP) (cit. on pp. 98, 134).
— (Dec. 2004). Physical layer on the radio path; General description. TS 05.01. 3rd Generation
Partnership Project (3GPP) (cit. on pp. 96, 97).
— (Feb. 2005a). Channel coding. TS 05.03. 3rd Generation Partnership Project (3GPP)
(cit. on p. 98).
— (Nov. 2005b). Radio Transmission and Reception. TS 05.05. 3rd Generation Partnership
Project (3GPP) (cit. on p. 96).
3GPP website. 3GPP - Specifications. url: http://www.3gpp.org/specifications
(visited on 08/11/2012) (cit. on p. 29).
3GPP2 (Dec. 1999). C.S0010-0 Recommended Minimum Performance Standards for cdma2000
Spread Spectrum Base Stations (cit. on p. 136).
— (Feb. 2004a). C.S0002-D Physical Layer Standard for cdma2000 Spread Spectrum Systems
(cit. on pp. 136, 138).
— (Feb. 2004b). C.S0005-D Upper Layer (Layer 3) Signaling Standard for cdma2000 Spread
Spectrum Systems (cit. on p. 167).
— (Mar. 2004c). C.S0022-A Position Determination Service for cdma2000 Spread Spectrum
Systems (cit. on p. 157).
Abate, J. and W. Whitt (Jan. 1992). “The Fourier-series method for inverting transforms
of probability distributions”. In: Queueing Systems 10 (1-2), pp. 5–87 (cit. on p. 66).
Abdi, A., K. Wills, H. Barger, M.-S. Alouini and M. Kaveh (2000). “Comparison of the
level crossing rate and average fade duration of Rayleigh, Rice and Nakagami fading
models with mobile channel data”. In: Vehicular Technology Conference, 2000. IEEE
VTS-Fall VTC 2000. 52nd. Vehicular Technology Conference, 2000. IEEE VTS-Fall
VTC 2000. 52nd. Vol. 4, 1850–1857 vol.4 (cit. on p. 70).
Agarwal, N. et al. (Dec. 2002). “Algorithms for GPS operation indoors and downtown”.
In: GPS Solutions 6 (3), pp. 149–160 (cit. on p. 170).
Akopian, D. and J. Syrjärinne (Oct. 2009). “A Fast Positioning Method Without Navi-
gation Data Decoding for Assisted GPS Receivers”. In: Vehicular Technology, IEEE
Transactions on 58 (8), pp. 4640 –4645 (cit. on p. 132).
Alexander, S. (1983). “Characterising buildings for propagation at 900 MHz”. In:
Electronics Letters 19 (20), p. 860 (cit. on p. 18).
189
References
Allan, D. (1966). “Statistics of atomic frequency standards”. In: Proceedings of the IEEE
54 (2), pp. 221–230 (cit. on p. 102).
Anghileri, M., M. Paonni, S. Wallner, J.-A. Ávila-Rodríguez and B. Eissfeller (Jan. 2010).
“Performance Assessment of GNSS Signals in Terms of Time to First Fix for Cold,
Warm and Hot Start”. In: Proceedings of the 2010 International Technical Meeting of The
Institute of Navigation. San Diego, CA, pp. 1051–1066 (cit. on pp. 68, 86).
Axelrad, P., B. K. Bradley, J. Donna, M. Mitchell and S. Mohiuddin (2011). “Collective
Detection and Direct Positioning Using Multiple GNSS Satellites”. In: Navigation 58
(4), pp. 305–321 (cit. on pp. 52, 175, 179).
Bahl, P. and V. Padmanabhan (2000). “RADAR: An in-building RF-based user location
and tracking system”. In: INFOCOM 2000. Nineteenth Annual Joint Conference of
the IEEE Computer and Communications Societies. Proceedings. IEEE. Vol. 2, 775–784
(cit. on p. 19).
Bastide, F., O. Julien, C. Macabiau and B. Roturier (Feb. 2001). “Analysis of L5/E5
Acquisition, Tracking and Data Demodulation Thresholds”. In: Proceedings of the
15th International Technical Meeting of the Satellite Division of The Institute of
Navigation (ION GPS 2002), pp. 2196–2207 (cit. on p. 61).
Berger, J. O. (Aug. 1985). Statistical Decision Theory and Bayesian Analysis. 2nd. Springer
(cit. on pp. 37, 44).
Biacs, Z., G. Marshall, M. Moeglein and W. Riley (Sept. 2002). “The Qualcomm/SnapTrack
Wireless-Assisted GPS Hybrid Positioning System and Results from Initial Com-
mercial Deployments”. In: Proceedings of the 15th International Technical Meeting
of the Satellite Division of The Institute of Navigation (ION GPS 2002), pp. 378–384
(cit. on p. 167).
Blom, H. and Y. Bar-Shalom (1988). “The interacting multiple model algorithm for sys-
tems with Markovian switching coefficients”. In: Automatic Control, IEEE Transactions
on 33 (8), pp. 780–783 (cit. on p. 118).
Borio, D., L. Camoriano, L. Lo Presti and M. Fantino (2008a). “DTFT-based frequency lock
loop for GNSS applications”. In: Aerospace and Electronic Systems, IEEE Transactions
on 44 (2), pp. 595–612 (cit. on p. 182).
Borio, D., L. Camoriano and L. Lo Presti (2008b). “Impact of GPS acquisition strategy
on decision probabilities”. In: Aerospace and Electronic Systems, IEEE Transactions on
44 (3), pp. 996–1011 (cit. on p. 71).
Borre, K., D. Akos, N. Bertelsen, P. Rinder and S. Jensen (2007). A Software-Defined GPS
and Galileo Receiver. Birkhäuser Boston (cit. on p. 50).
Bregni, S. and L. Barbieri (Dec. 2003). “Experimental evaluation of the impact of network
frequency synchronization on GSM quality of service during handover”. In: Global
Telecommunications Conference, 2003. GLOBECOM ’03. IEEE. Vol. 6, 3457 –3461 vol.6
(cit. on p. 99).
Broumandan, A., T. Lin, J. Nielsen and G. Lachapelle (Sept. 2008). “Practical Results of
Hybrid AOA/TDOA Geo-Location Estimation in CDMA Wireless Networks”. In:
Vehicular Technology Conference, 2008. VTC 2008-Fall. IEEE 68th. Vehicular Technology
Conference, 2008. VTC 2008-Fall. IEEE 68th, pp. 1 –5 (cit. on p. 175).
Brown, K. R. J. (Sept. 1991). “The Theory of the GPS Composite Clock”. In: Proceedings
of the 4th International Technical Meeting of the Satellite Division of The Institute of
Navigation. ION GPS 1991. Albuquerque, NM, pp. 223–242 (cit. on p. 124).
190
References
Brown, R. G. and P. Y. C. Hwang (1997). Introduction to Random Signals and Applied
Kalman Filtering with MATLAB Exercises. Vol. 3rd ed. John Wiley & Sons, Inc. (cit. on
pp. 42, 43, 103–105).
Caffery, J. J. and G. Stuber (May 1998a). “Subscriber location in CDMA cellular networks”.
In: IEEE Transactions on Vehicular Technology 47 (2), pp. 406 –416 (cit. on pp. 137, 159).
Caffery, J. and G. Stuber (Apr. 1998b). “Overview of radiolocation in CDMA cellular
systems”. In: Communications Magazine, IEEE 36 (4), pp. 38 –45 (cit. on p. 31).
Cavers, J. K. (1978). “On the Fast Fourier Transform Inversion of Probability Generating
Functions”. In: IMA Journal of Applied Mathematics 22 (3), pp. 275–282 (cit. on pp. 66,
67).
Chaffee, J. and J. Abel (Apr. 1994). “GDOP and the Cramer-Rao bound”. In: , IEEE
Position Location and Navigation Symposium, 1994. , IEEE Position Location and
Navigation Symposium, 1994, pp. 663 –668 (cit. on p. 52).
Chaffee, J. (Nov. 1987). “Relating the Allan Variance to the Diffusion Coefficients
of a Linear Stochastic Differential Equation Model for Precision Oscillators”. In:
Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on 34 (6), pp. 655
–658 (cit. on p. 105).
Closas, P., C. Fernandez-Prades and J. Fernandez-Rubio (2007). “Maximum Likelihood
Estimation of Position in GNSS”. In: Signal Processing Letters, IEEE 14 (5), pp. 359–362
(cit. on pp. 52, 175, 179).
Cong, L. and W. Zhuang (Mar. 2004). “Nonline-of-sight error mitigation in mobile
location”. English. In: INFOCOM 2004. Twenty-third AnnualJoint Conference of the IEEE
Computer and Communications Societies. INFOCOM 2004. Twenty-third AnnualJoint
Conference of the IEEE Computer and Communications Societies. Vol. 1. IEEE,
p. 659 (cit. on pp. 159, 160).
Corazza, G. E, C. Caini, A. Vanelli-Coralli and A. Polydoros (July 2004). “DS-CDMA
code acquisition in the presence of correlated fading - Part I: theoretical aspects”.
English. In: IEEE Transactions on Communications 52 (7), pp. 1160–1168 (cit. on p. 91).
Corazza, G. (1996). “On the MAX/TC criterion for code acquisition and its application to
DS-SSMA systems”. In: Communications, IEEE Transactions on 44 (9), pp. 1173–1182
(cit. on p. 71).
Couronneau, N., P. Duffett-Smith and A. Mitelman (Jan. 2011). “Acquisition Time
of the GPS L1 C/A Signal in an Indoor Environment”. In: Proceedings of the 2011
International Technical Meeting of The Institute of Navigation. San Diego, CA, pp. 767
–774 (cit. on p. 60).
Dardari, D., A. Conti, U. Ferner, A. Giorgetti and M. Win (Feb. 2009). “Ranging With
Ultrawide Bandwidth Signals in Multipath Environments”. In: Proceedings of the
IEEE 97 (2), pp. 404 –426 (cit. on p. 41).
Davenport, R. (1991). “FFT processing of direct sequence spreading codes using modern
DSP microprocessors”. In: Aerospace and Electronics Conference, 1991. NAECON 1991.,
Proceedings of the IEEE 1991 National. Aerospace and Electronics Conference, 1991.
NAECON 1991., Proceedings of the IEEE 1991 National, 98–105 vol.1 (cit. on pp. 48,
182).
Davis, J. A., C. A. Greenhall and P. W. Stacey (Feb. 2005). “A Kalman filter clock algorithm
for use in the presence of flicker frequency modulation noise”. In: Metrologia 42 (1),
pp. 1–10 (cit. on p. 104).
191
References
Dempster, A. P., N. M. Laird and D. B. Rubin (Jan. 1977). “Maximum Likelihood from
Incomplete Data via the EM Algorithm”. In: Journal of the Royal Statistical Society.
Series B (Methodological) 39 (1). ArticleType: research-article / Full publication date:
1977 / Copyright © 1977 Royal Statistical Society, pp. 1–38 (cit. on p. 139).
DiCarlo, D. and C. Weber (Aug. 1980). “Statistical Performance of Single Dwell Serial
Synchronization Systems”. In: Communications, IEEE Transactions on 28 (8), pp. 1382
–1388 (cit. on p. 68).
Dietrich C.B., J., K. Dietze, J. Nealy and W. Stutzman (Sept. 2001). “Spatial, polarization,
and pattern diversity for wireless handheld terminals”. In: IEEE Transactions on
Antennas and Propagation 49 (9), pp. 1271 –1281 (cit. on p. 174).
Diggelen, F van (Jan. 2007). “GNSS Accuracy – Lies, Damn Lies and Statistics”. In: GPS
World 18 (1) (cit. on p. 174).
Dow, J. M., R. E. Neilan and C. Rizos (Feb. 2009). “The International GNSS Service in a
changing landscape of Global Navigation Satellite Systems”. In: Journal of Geodesy
83 (3-4), pp. 191–198 (cit. on pp. 109, 168).
Drane, C., M. Macnaughtan and C. Scott (Apr. 1998). “Positioning GSM telephones”.
In: IEEE Communications Magazine 36 (4), pp. 46 –54, 59 (cit. on p. 31).
Duffett-Smith, P. and R. Rowe (June 2001). “Comparative A-GPS and 3G-Matrix Testing
in a Dense Urban Environment”. In: Proceedings of the 19th International Technical
Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2006),
pp. 3006–3011 (cit. on p. 31).
Duffett-Smith, P. J and J. Craig (2004). “Matrix, and enhanced satellite positioning”.
English. In: Fifth IEE International Conference on 3G Mobile Communication Technologies,
2004. 3G 2004. Fifth IEE International Conference on 3G Mobile Communication
Technologies, 2004. 3G 2004. IET, pp. 364–367 (cit. on pp. 31, 133).
Duffett-Smith, P. J, P. Hansen, S. D Hern and A. R Pratt (Apr. 2007). “Atomic Quality
Clocks for Cell-Phone GPS without the Atoms”. In: Proceedings of the 63rd Annual
Meeting of The Institute of Navigation, pp. 496–503 (cit. on p. 96).
Duffett-Smith, P. J. and G. Woan (1992). “The CURSOR Radio Navigation and Tracking
System”. In: The Journal of Navigation 45 (02), pp. 157–165 (cit. on p. 18).
Duffett-Smith, P. and M. Macnaughtan (May 2002). “Precise UE positioning in UMTS
using cumulative virtual blanking”. In: 3G Mobile Communication Technologies, 2002.
Third International Conference on (Conf. Publ. No. 489). 3G Mobile Communication
Technologies, 2002. Third International Conference on (Conf. Publ. No. 489), pp. 355
–359 (cit. on p. 144).
Elnahrawy, E., X. Li and R. Martin (Oct. 2004). “The limits of localization using signal
strength: a comparative study”. In: 2004 First Annual IEEE Communications Society
Conference on Sensor and Ad Hoc Communications and Networks, 2004. IEEE SECON
2004. 2004 First Annual IEEE Communications Society Conference on Sensor and
Ad Hoc Communications and Networks, 2004. IEEE SECON 2004, pp. 406 –414
(cit. on p. 19).
ETSI website. ETSI GSM. url: http://www.etsi.org/website/technologies/gsm.
aspx (visited on 08/11/2012) (cit. on p. 29).
European Union (Sept. 2012). European GNSS (Galileo) Open Service Signal In Space
Interface Control Document (cit. on p. 27).
192
References
Faragher, R. (Aug. 2007). “Studies of radio propagation in urban environments and
the effects of multipath interference on positioning systems”. Cambridge, UK:
University of Cambridge (cit. on pp. 95, 96, 105, 116, 139).
Faragher, R. and P. Duffett-Smith (Dec. 2010). “Measurements of the effects of multipath
interference on timing accuracy in a cellular radio positioning system”. In: Radar,
Sonar Navigation, IET 4 (6), pp. 818 –824 (cit. on pp. 116, 148).
Faragher, R., C. Sarno and M. Newman (Apr. 2012). “Opportunistic radio SLAM for
indoor navigation using smartphone sensors”. In: Position Location and Navigation
Symposium (PLANS), 2012 IEEE/ION. Position Location and Navigation Symposium
(PLANS), 2012 IEEE/ION, pp. 120 –128 (cit. on p. 22).
Favey, E., D. Megnet, A. Rempfler and H. Mathis (May 2001). “Integration of a
GSM Receiver with GPS for Integrated Navigation or for Quick Satellite Signal
Acquisition”. In: Proceedings of the 18th International Technical Meeting of the
Satellite Division of The Institute of Navigation (ION GNSS 2005), pp. 1425–1432
(cit. on pp. 95, 115).
Fessler, J. and A. Hero (Oct. 1994). “Space-alternating generalized expectation-maximiz-
ation algorithm”. In: IEEE Transactions on Signal Processing 42 (10), pp. 2664 –2677
(cit. on p. 139).
Fischer, S., H. Grubeck, A. Kangas, H. Koorapaty, E. Larsson and P. Lundqvist (Sept.
1998). “Time of arrival estimation of narrowband TDMA signals for mobile posi-
tioning”. English. In: The Ninth IEEE International Symposium on Personal, Indoor and
Mobile Radio Communications, 1998. The Ninth IEEE International Symposium on
Personal, Indoor and Mobile Radio Communications, 1998. Vol. 1. IEEE, 451–455
vol.1 (cit. on p. 158).
Fleury, B., M. Tschudin, R. Heddergott, D. Dahlhaus and K. Ingeman Pedersen (Mar.
1999). “Channel parameter estimation in mobile radio environments using the SAGE
algorithm”. In: IEEE Journal on Selected Areas in Communications 17 (3), pp. 434–450
(cit. on p. 139).
Galleani, L and P Tavella (June 2003). “On the use of the Kalman filter in timescales”.
In: Metrologia 40 (3), S326–S334 (cit. on p. 124).
Gold, R. (1967). “Optimal binary sequences for spread spectrum multiplexing (Corresp.)”
In: Information Theory, IEEE Transactions on 13 (4), pp. 619–621 (cit. on p. 26).
Google Maps (Dec. 2012). url: https://maps.google.com/ (visited on 13/12/2012)
(cit. on p. 161).
GPS Joint Program Office (Dec. 2004). Navstar Global Positioning System Interface Specifi-
cation (cit. on p. 27).
Groves, P. D. (2011). “Shadow Matching: A New GNSS Positioning Technique for Urban
Canyons”. In: The Journal of Navigation 64 (03), pp. 417–430 (cit. on p. 21).
GSM Association website. GSM | About Us. url: http://www.gsma.com/aboutus/gsm-
technology/gsm/ (visited on 08/11/2012) (cit. on p. 29).
Gumacos, C. (Mar. 1963). “Analysis of an Optimum Sync Search Procedure”. English.
In: IEEE Transactions on Communications Systems 11 (1), pp. 89–99 (cit. on p. 62).
Gustafsson, F. and F. Gunnarsson (July 2005). “Mobile positioning using wireless
networks: possibilities and fundamental limitations based on available wireless
network measurements”. In: IEEE Signal Processing Magazine 22 (4), pp. 41 –53
(cit. on p. 30).
Hann, K. (Mar. 2012). LTE Stretches Synchronization to New Limis (cit. on p. 100).
193
References
Hashemi, H. (July 1993). “The indoor radio propagation channel”. In: Proceedings of the
IEEE 81 (7), pp. 943 –968 (cit. on p. 20).
Holmes, J. and Chang Chen (1977). “Acquisition Time Performance of PN Spread-
Spectrum Systems”. In: Communications, IEEE Transactions on 25 (8), pp. 778–784
(cit. on pp. 59, 62, 64, 67, 86).
Holmes, J. K., N. Morgan and P. Dafesh (Sept. 2006). “Theoretical Approach to De-
termining the 95% Probability of TTFF for the P(Y) Code Utilizing Active Code
Acquisition”. In: Proceedings of the 19th International Technical Meeting of the Satellite
Division of The Institute of Navigation. ION GNSS 2006. Fort Worth, TX, pp. 201–211
(cit. on pp. 66, 68, 78).
Hopkins, P. (Aug. 1977). “A Unified Analysis of Pseudonoise Synchronization by
Envelope Correlation”. English. In: IEEE Transactions on Communications 25 (8),
pp. 770–778 (cit. on pp. 59, 62).
“IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and
Time Metrology—Random Instabilities” (2008). In: IEEE STD 1139-2008, pp. c1 –35
(cit. on p. 103).
Jan, S.-S. and Y.-C. Lin (2009). “A new multi-C/A code acquisition method for GPS”. In:
GPS Solutions 13 (4), pp. 293–303 (cit. on p. 84).
Jaynes, E. T. (June 2003). Probability Theory: The Logic of Science. Ed. by G. L. Bretthorst.
Cambridge University Press. 758 pp. (cit. on p. 35).
Jones, E., T. Oliphant, P. Peterson and others (2001). SciPy: Open source scientific tools for
Python (cit. on p. 117).
Ju-Yong Do, M. Rabinowitz and P. Enge (July 2007). “Robustness of TOA and TDOA
Positioning Under Suboptimal Weighting Conditions”. English. In: IEEE Transactions
on Aerospace and Electronic Systems 43 (3), pp. 1177–1180 (cit. on p. 50).
Kalman, R. E. (1960). “A New Approach to Linear Filtering and Prediction Problems”.
In: Transactions of the ASME–Journal of Basic Engineering 82 (Series D), 35–45 (cit. on
pp. 23, 42).
Kasper, J. and C. Hutchinson (May 1978). “The Omega navigation system–An overview”.
In: IEEE Communications Society Magazine 16 (3), pp. 23 –35 (cit. on p. 23).
Kay, S. M. (1993). Fundamentals of statistical signal processing: estimation theory. Upper
Saddle River, NJ, USA: Prentice-Hall, Inc. (cit. on pp. 35, 40, 41).
Kim, W., G.-I. Jee and J. Lee (2001). “Wireless location with NLOS error mitigation in
Korean CDMA system”. In: 3G Mobile Communication Technologies, 2001. Second Inter-
national Conference on (Conf. Publ. No. 477). 3G Mobile Communication Technologies,
2001. Second International Conference on (Conf. Publ. No. 477), pp. 134 –138 (cit. on
p. 159).
Klukas, R. (Sept. 1997). “Cellular Telephone Positioning Using GPS Time Synchroniza-
tion”. In: Proceedings of the 10th International Technical Meeting of the Satellite
Division of The Institute of Navigation (ION GPS 1997), pp. 1405–1414 (cit. on
pp. 30, 140).
Lannelongue, S. and P. Pablos (June 1998). “Fast Acquisition Techniques For G.P.S.
Receivers”. In: Proceedings of the 54th Annual Meeting of The Institute of Navigation.
Denver, CO, pp. 261 –269 (cit. on p. 54).
Leon, F. de and J. Marciano (Nov. 2006). “Application of MUSIC, ESPRIT and SAGE
Algorithms for Narrowband Signal Detection and Localization”. In: TENCON 2006.
194
References
2006 IEEE Region 10 Conference. TENCON 2006. 2006 IEEE Region 10 Conference,
pp. 1 –4 (cit. on p. 140).
Li, W., W. Yao and P. Duffett-Smith (Jan. 2009). “Comparative Study of Joint TOA/DOA
Estimation Techniques for Mobile Positioning Applications”. In: 6th IEEE Consumer
Communications and Networking Conference, 2009. CCNC 2009, pp. 1 –5 (cit. on p. 140).
Lohan, E.-S., A. Lakhzouri and M. Renfors (2004). “Selection of the multiple-dwell
hybrid-search strategy for the acquisition of Galileo signals in fading channels”.
In: 2004 IEEE 15th International Symposium on Personal, Indoor and Mobile Radio
Communications (IEEE Cat. No.04TH8754). 2004 IEEE 15th International Symposium
on Personal, Indoor and Mobile Radio Communications. Barcelona, Spain, pp. 2352–
2356 (cit. on p. 62).
Malte Spitz. Malte Spitz: Your phone company is watching | Video on TED.com (cit. on p. 30).
Mehra, R. (Apr. 1970). “On the identification of variances and adaptive Kalman filtering”.
English. In: IEEE Transactions on Automatic Control 15 (2), pp. 175–184 (cit. on p. 116).
Mensing, C., S. Sand and A. Dammann (2010). “Hybrid Data Fusion and Tracking for
Positioning with GNSS and 3GPP-LTE”. In: International Journal of Navigation and
Observation 2010, pp. 1–12 (cit. on p. 167).
Meyr, H. and G. Polzer (Dec. 1983). “Performance Analysis for General PN-Spread-
Spectrum Acquisition Techniques”. In: Communications, IEEE Transactions on 31 (12),
pp. 1317 –1319 (cit. on p. 68).
Mitelman, A., P.-L. Normark, M. Reidevall, J. Thor, O. Gröqvist and L. Magnusson
(Sept. 2006). “The Nordnav Indoor GNSS Reference Receiver”. In: Proceedings of the
19th International Technical Meeting of the Satellite Division of The Institute of Navigation.
ION GNSS 2006. Fort Worth, TX, pp. 2494–2502 (cit. on pp. 79, 106).
Morley, C. (2003). XmlRpc++ (cit. on p. 107).
Murphy, K. P. (1998). Switching Kalman Filters (cit. on p. 118).
Nakagami, M (1960). “The m-distribution – A general formula of intensity distribution
of rapid fading”. In: Statistical Methods in Radio Wave Propagation. Ed. by W. Hoffmann
(cit. on p. 149).
National Coordination Office for Space-Based Positioning, Navigation, and Timing
(Sept. 2012). GPS.gov. url: http://www.gps.gov/ (visited on 15/09/2012) (cit. on
p. 24).
Neven, W., T. Quilter, R. Weedon and R. Hogendoorn (Aug. 2005). Report on Wide Area
Multilateration EATMP TRS 131/04 (cit. on p. 18).
Newman, P. et al. (Nov. 2009). “Navigating, Recognizing and Describing Urban Spaces
With Vision and Lasers”. In: Int. J. Rob. Res. 28 (11-12), 1406–1433 (cit. on p. 22).
Nuttall, A. (1970). “Alternate forms for numerical evaluation of cumulative probability
distributions directly from characteristic functions”. In: Proceedings of the IEEE 58
(11), pp. 1872–1873 (cit. on p. 66).
O’Driscoll, C., M. Petovello and G. Lachapelle (Jan. 2008). “Software Receiver Strategies
for the Acquisition and Re-Acquisition of Weak GPS Signals”. In: Proceedings of ION
NTM08 (cit. on p. 86).
O’Driscoll, C. (2007). “Performance Analysis of the Parallel Acquisition of Weak GPS
Signals”. Department of Electrical and Electronic Engineering, National University
of Ireland (cit. on pp. 62, 66, 72).
Ofcom (2012). Ofcom | Facts & Figures. url: http://media.ofcom.org.uk/facts/
(visited on 16/09/2012) (cit. on p. 28).
195
References
O’Mahony, N., G. Lachapelle and C. C. Murphy (2011). “Simplifying the Performance
Analysis of the SPRT for GPS Acquisition”. In: International Journal of Navigation and
Observation 2011, pp. 1–15 (cit. on p. 62).
Pany, T. (2010). Navigation Signal Processing for GNSS Software Receivers. 1st ed. Artech
House Publishers. 330 pp. (cit. on pp. 35, 39–41, 48, 72, 105).
Parkinson, B. W. and J. J. Spilker (Jan. 1996). Global Positioning System: Theory & Applica-
tions (Volume One). 1st. (AIAA) American Institute of Aeronautics & Astronautics
(cit. on pp. 24, 26, 49, 50, 70, 147).
Patel, P. and J. Holtzman (June 1994). “Analysis of a simple successive interference
cancellation scheme in a DS/CDMA system”. In: IEEE Journal on Selected Areas in
Communications 12 (5), pp. 796 –807 (cit. on p. 142).
Pauluzzi, D. and N. Beaulieu (Oct. 2000). “A comparison of SNR estimation techniques
for the AWGN channel”. In: IEEE Transactions on Communications 48 (10), pp. 1681–
1691 (cit. on p. 80).
Pi-Chun Chen (1999). “A non-line-of-sight error mitigation algorithm in location esti-
mation”. English. In: 1999 IEEE Wireless Communications and Networking Conference,
1999. WCNC. 1999 IEEE Wireless Communications and Networking Conference,
1999. WCNC. IEEE, 316–320 vol.1 (cit. on pp. 158, 160).
Polydoros, A. and C. Weber (1984a). “A Unified Approach to Serial Search Spread-
Spectrum Code Acquisition–Part I: General Theory”. In: IEEE Transactions on
Communications 32 (5), pp. 542–549 (cit. on pp. 62, 64, 66, 75, 86).
— (1984b). “A Unified Approach to Serial Search Spread-Spectrum Code Acquisition–
Part II: A Matched-Filter Receiver”. In: Communications, IEEE Transactions on 32 (5),
pp. 550–560 (cit. on p. 62).
Proakis, J. (Aug. 2000). Digital Communications. 4th ed. McGraw-Hill Science/Engineer-
ing/Math (cit. on p. 37).
“Radio positioning systems” (Mar. 2003). Pat. 6529165. P. J. Duffett-Smith, J. P. Brice
and P. Hansen (cit. on p. 18).
“Radio positioning systems” (Apr. 2008). Pat. 7359719. P. J. Duffett-Smith, J. P. Brice and
P. Hansen. U.S. Classification: 455/456.6 (cit. on p. 18).
Rasmussen, C. E. (2006). “Gaussian processes for machine learning”. In: MIT Press
(cit. on p. 127).
Riley, K. F., M. P. Hobson and S. J. Bence (2006). Mathematical methods for physics and
engineering. Cambridge University Press. 1362 pp. (cit. on p. 64).
Ristic, B., S. Arulampalam and N. Gordon (2004). Beyond the Kalman Filter: Particle Filters
for Tracking Applications. en. Artech House. 326 pp. (cit. on p. 43).
Russian Institute of Space Device Engineering (2008). Global Navigation Sattelite System
GLONASS Interface Control Document (cit. on p. 27).
Saleh, A. and R. Valenzuela (Feb. 1987). “A Statistical Model for Indoor Multipath
Propagation”. In: IEEE Journal on Selected Areas in Communications 5 (2), pp. 128 –137
(cit. on p. 149).
Schmidt, R. (Mar. 1986). “Multiple emitter location and signal parameter estimation”.
In: IEEE Transactions on Antennas and Propagation 34 (3), pp. 276 –280 (cit. on p. 30).
Simon, M. K., M.-S. Alouini and S. Member (1998). “A Unified Approach to the Perfor-
mance Analysis of Digital Communication over Generalized Fading Channels”. In:
Proc. IEEE, 1860–1877 (cit. on pp. 70, 90).
196
References
Sirola, N. and P. Syrjärinne (2002). “Solving GPS time and position without navigation
data”. In: Proceedings of the ENC-GNSS 2002 (cit. on p. 54).
Stirling-Gallacher, R., A. Hulbert and G. Povey (1996). “A fast acquisition technique for
a direct sequence spread spectrum signal in the presence of a large Doppler shift”. In:
Spread Spectrum Techniques and Applications Proceedings, 1996., IEEE 4th International
Symposium on. Spread Spectrum Techniques and Applications Proceedings, 1996.,
IEEE 4th International Symposium on. Vol. 1, 156–160 vol.1 (cit. on p. 84).
Syrjärinne, J. (2000). “Time recovery through fusion of inaccurate network timing
assistance with GPS measurements”. In: Information Fusion, 2000. FUSION 2000.
Proceedings of the Third International Conference on. Information Fusion, 2000. FUSION
2000. Proceedings of the Third International Conference on. Vol. 2, WED5/3–WED510
vol.2 (cit. on p. 132).
Thrun, S. et al. (June 2006). “Stanley: The Robot that Won the DARPA Grand Challenge”.
In: Journal of Field Robotics 23 (9). Ed. by W. Periodicals, pp. 661–692 (cit. on p. 22).
“Transfer of calibrated time information in a mobile terminal” (Dec. 2010). Pat. 7852267.
P. J. Duffett-Smith, A. R. Pratt and D. W. Bartlett. Granted (cit. on p. 96).
Trevisani, E. and A. Vitaletti (Dec. 2004). “Cell-ID location technique, limits and benefits:
an experimental study”. In: Sixth IEEE Workshop on Mobile Computing Systems and
Applications, 2004. WMCSA 2004. Sixth IEEE Workshop on Mobile Computing
Systems and Applications, 2004. WMCSA 2004, pp. 51 –60 (cit. on p. 20).
United States Coast Guard (1994). Specification of the transmitted Loran-C signal. Dis-
tributed to depository libraries in microfiche. Shipping list no.: 94-0699-M. Wash-
ington, DC: U.S. Dept. of Transportation, U.S. Coast Guard (cit. on p. 23).
Van Dierendonck, A. J, J. B McGraw and R. G Brown (Nov. 1984). Relationship Between
Allan Variances and Kalman Filter Parameters (cit. on pp. 103–105).
van Diggelen, F. (2002). “Indoor GPS theory & implementation”. In: Position Location and
Navigation Symposium, 2002 IEEE. Position Location and Navigation Symposium,
2002 IEEE, pp. 240–247 (cit. on pp. 61, 82).
— (Mar. 2009). A-GPS: Assisted GPS, GNSS, and SBAS. 1st ed. Artech House (cit. on
pp. 50, 53, 54).
Van Trees, H. L. (1968). Detection, estimation, and modulation theory. New York: Wiley
(cit. on pp. 35, 44).
Ward, P. W., J. W. Betz and C. J. Hegarty (2006). “Satellite Signal Acquisition, Tracking,
and Data Demodulation”. In: Understanding GPS: Principles and Applications. 2nd
edition. Chapter 5: Artech House (cit. on p. 84).
Wax, M. and T. Kailath (Apr. 1985). “Detection of signals by information theoretic
criteria”. In: IEEE Transactions on Acoustics, Speech and Signal Processing 33 (2), pp. 387
–392 (cit. on p. 142).
Weiss, A. and E. Weinstein (Apr. 1983). “Fundamental limitations in passive time delay
estimation–Part I: Narrow-band systems”. In: IEEE Transactions on Acoustics, Speech
and Signal Processing 31 (2), pp. 472 –486 (cit. on p. 41).
Wylie-Green, M. P and S. S Wang (2001). “Robust range estimation in the presence of
the non-line-of-sight error”. English. In: Vehicular Technology Conference, 2001. VTC
2001 Fall. IEEE VTS 54th. Vehicular Technology Conference, 2001. VTC 2001 Fall.
IEEE VTS 54th. Vol. 1. IEEE, 101–105 vol.1 (cit. on pp. 158, 160).
197
References
Yong-Hwan Lee and Seung-Jun Kim (Nov. 2000). “Sequence acquisition of DS-CDMA
systems employing Gold sequences”. English. In: IEEE Transactions on Vehicular
Technology 49 (6), pp. 2397–2404 (cit. on p. 62).
Ziedan, N. I. (2006). GNSS Receivers for Weak Signals. Boston, Mass: Artech House.
234 pp. (cit. on p. 170).
Ziv, J. and M. Zakai (May 1969). “Some lower bounds on signal parameter estimation”.
In: IEEE Transactions on Information Theory 15 (3), pp. 386 –391 (cit. on p. 41).
Zucca, C. and P. Tavella (Feb. 2005). “The clock model and its relationship with the
Allan and related variances”. In: Ultrasonics, Ferroelectrics and Frequency Control,
IEEE Transactions on 52 (2), pp. 289 –296 (cit. on pp. 104, 105).
198