Master of Science Thesis

Transcription

TAMPERE UNIVERSITY OF TECHNOLOGY
Degree program in Information Technology
A.K.M.NAJMUL ISLAM
CNR ESTIMATION AND INDOOR CHANNEL MODELING OF GPS
SIGNALS
Master of Science Thesis
Examiners: Docent Elena-Simona Lohan
and Prof. Markku Renfors
Examiners and topic approved in
the council meeting of the Faculty of
Computing and Electrical Engineering
on the 16th Jan, 2008
Abstract
TAMPERE UNIVERSITY OF TECHNOLOGY
Degree Program in Information Technology, Department of Communications
Engineering
ISLAM, A.K.M.NAJMUL: CNR estimation and indoor channel modeling of GPS
signals
Master of Science Thesis, 68 pages, 3 appendix pages
March, 2008
Major: Communications Engineering
Examiners: Dr. Elena-Simona Lohan, Prof. Markku Renfors
Keywords: Binary Offset Carrier, Binary Phase Shift Keying, Carrier to Noise Ratio,
Global Positioning System, Galileo, Pseudolites, Satellites
In recent studies, accurate positioning of terminals has received much attention in
wireless communications research. One reason is that of the requirement for emergency
call positioning imposed by the authorities. The positioning algorithms based on the
Global Positioning System (GPS) have limitations in the indoor environments because
the signal experiences severe attenuation in such situations.
Estimation of the Carrier to Noise Ratio (CNR) is one of the most important functionalities of the Global Navigation Satellite Systems (GNSSs) receivers. However, the
conventional GPS receivers are not able to estimate the CNR accurately enough in moderate or severe indoor reception. In this thesis, several moment-based CNR estimators
are derived and the author shows the results for both Binary Offset Carrier (BOC) and
Binary Phase Shift Keying (BPSK) modulated signals. BOC modulation is to be used in
modernized GPS signals and for Galileo, the European navigation system, while BPSK
is currently employed by basic GPS signals. The results of different estimators are compared in order to find the most robust estimator.
On the other hand, the indoor propagation characteristics of the GPS signals are required to be well understood in order to derive good navigation algorithms suitable for
indoor environments. In this thesis, the indoor propagation channel using pseudolites and
satellites are also analyzed, based on the measurement data collected in different scenarios.
i
Preface
I have written this Master of Science thesis for the Department of Communications Engineering, Tampere University of Technology, Finland. I have done the work for this thesis
at the Department of Communications Engineering under the projects, Advanced Techniques for Personal Navigation (ATENA) and Future GNSS Applications and Techniques
(FUGAT) funded by the Finnish Funding Agency for Technology and Innovation (Tekes)
and some participating companies. I would like to express my gratitude to my thesis supervisors Dr. Elena Simona Lohan and Prof. Markku Renfors for their valuable guidance
and assistance during the thesis work. I would also like to thank Dr. Yuan Yang, Danai
Skournetou, and Hu Xuan for their friendly support during the work. Finally, I express
my gratitude to my parents for their endless love and inspiration.
This endeavor is dedicated to my wife, Nasreen Azad.
Tampere, Finland.
25th March, 2008
A.K.M. Najmul Islam
Insinöörinkatu 60 B 86
33720 TAMPERE
najmul.islam(at)tut.fi
Tel. int. +358 50 934 2886
ii
Contents
Abstract
i
Preface
ii
List of Abbreviations
vi
List of Symbols
1
2
3
Introduction
1.1 Motivation for the research topic
1.2 Objective of the thesis . . . . . .
1.3 Thesis contributions . . . . . . .
1.4 Thesis outline . . . . . . . . . .
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Overview of GPS and Galileo systems
3.1 Basic GPS overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Fading channel modeling overview
2.1 Multi-path channel parameters . . . . . .
2.1.1 Time dispersion parameters . . .
2.1.2 Coherence bandwidth . . . . . . .
2.1.3 Doppler shift and Doppler spread
2.1.4 Coherence time . . . . . . . . . .
2.2 Fading channel models . . . . . . . . . .
2.2.1 Rician fading channel . . . . . .
2.2.2 Rayleigh fading channel . . . . .
2.2.3 Log-normal fading channel . . . .
2.2.4 Loo fading channel . . . . . . . .
2.2.5 Nakagami fading channel . . . .
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CONTENTS
3.2
3.3
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iv
Modernized GPS system overview . . . . . . . . . . . . . . . . . . . . .
Galileo system overview . . . . . . . . . . . . . . . . . . . . . . . . . .
Measurement campaigns
4.1 Pseudolite based measurement campaign . . . .
4.1.1 Single pseudolite based measurement .
4.1.2 Multiple pseudolite based measurement
4.2 Satellite based measurement campaign . . . . .
Measurement data analysis
5.1 Acquisition of C/A code . . .
5.1.1 Search window . . . .
5.1.2 Search strategy . . . .
5.1.3 Correlation . . . . . .
5.2 Drift estimation . . . . . . . .
5.3 Navigation data bit estimation
5.4 Coherent integration . . . . .
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CNR Estimation
6.1 Signal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Coherent integration outputs . . . . . . . . . . . . . . . . . . . .
6.1.2 Non-coherent integration outputs . . . . . . . . . . . . . . . . .
6.1.3 PDFs and CDFs . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Moment-based CNR estimators . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 First-order moments (1STO) . . . . . . . . . . . . . . . . . . . .
6.2.2 Second-order moments, method 1 (2NDO-M1) . . . . . . . . . .
6.2.3 Second-order moments, method 2 (2NDO-M2) . . . . . . . . . .
6.2.4 Combined second (central) and first (non-central) order moments
(2NDO-1STO-M1) . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.5 Combined second (non-central) and first (central) order moments
(2NDO-1STO-M2) . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.6 Combined fourth and second order moment (4THO-2NDO) . . .
6.2.7 Combined fourth and first order moments (4THO-1STO) . . . . .
6.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Results for single path channel . . . . . . . . . . . . . . . . . . .
6.3.2 Results for multi-path channel . . . . . . . . . . . . . . . . . . .
6.3.3 Envelope vs. squared envelope as nonlinearity . . . . . . . . . .
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46
CONTENTS
6.4
6.5
6.6
6.7
7
8
CNR mappings . . . . . . . . . . . . . . . . .
CNR estimators results with measurement data
CNR estimation results for 4THO-1STO using
estimators . . . . . . . . . . . . . . . . . . . .
Computational complexity of the estimators . .
v
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different navigation bit
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Channel models based on measurement data
7.1 Proposed combined fading channel models . . . . .
7.2 Fading distribution matching . . . . . . . . . . . . .
7.2.1 Pseudolite results . . . . . . . . . . . . . . .
7.2.2 Satellite results . . . . . . . . . . . . . . . .
7.3 Average path number and time dispersion parameters
7.3.1 Pseudolite results . . . . . . . . . . . . . . .
7.3.2 Satellite results . . . . . . . . . . . . . . . .
7.4 Comparison between pseudolites and satellite results
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Conclusions and Future Works
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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References
69
Appendix A: Phase variation and Delay error estimation
74
List of Abbreviations
ATENA
Advanced Techniques for Personal Navigation
AWGN
Additive White Gaussian Noise
BOC
Binary Offset Carrier
BPSK
Binary Phase Shift Keying
C/A
Coarse/Acquisition code of GPS
CDF
Cumulative Distribution Function
CIR
Channel Impulse Response
CNR
Carrier-to-Noise Ratio
DS-SS
Direct sequence Spread Spectrum
E2-L1-E1
Frequency band centered in 1575.42 MHz
ESA
European Space Agency
FFT
Fast Fourier Transform
FUGAT
Future GNSS Applications and Techniques
GNSS
Global Navigation Satellite System
GPS
Global Positioning System
I
In-Phase
vi
LIST OF ABBREVIATIONS
IF
Intermediate Frequency
LOS
Line of Sight
MBOC
Multiplexed Binary Offset Carrier
MSE
Mean Square Error
NLOS
Non Line of Sight
OS
Open Service
PL
Pseudolite
PRN
Pseudo Random Number
PDF
Probability Distribution Function
Q
Quadrature
RHCP
Right Hand Circular Polarization
RMS
Root-Mean-Square
RMSE
Root-Mean-Square Error
RF
Radio Frequency
SNR
Signal-to-Noise Ratio
TUT
Tampere University of Technology
vii
List of Symbols
A
Envelope of the LOS delay
α
Complex channel coefficient
Bw
Bandwidth
c
Speed of Light
∆ fD
Doppler Error
∆τ
Delay Error
∆θ
Phase difference of two consecutive millisecond outputs
E(·)
Statistic average
Eb
Signal Power
Eeb
Signal Power in a correct bin
fc
Carrier frequency
fD
Maximum Doppler shift
fds
Doppler spread
fCombined−M1 ()
Combined distribution method-1
fCombined−M2 ()
fCombined−M3 ()
viii
List of Symbols
ix
fLogn ()
Lognormal distribution
fLoo ()
Loo distribution
fNaka ()
Nakagami distribution
fRayl ()
Rayleigh distribution
fRice ()
Rician distribution
KRice
Rician factor
I0
Modified 0th order Bessel function
n
Degree of freedom of chi-square distributed variable
m
Nakagami factor
µ
Mean
Nc
Coherent Integration length
Nnc
Non-coherent Integration length
N0
Noise Power
N ()
Normal Distribution
RBOC
BOC/BPSK autocorrelation function
σ2
Variance
CorridorSAT
Satellite data captured in office corridor
RoomSAT
Satellite data captured in office room
List of Symbols
x
MultiplePL
Data captured with multiple pseudolites
SinglePL
Data captured with single pseudolite
θn
Phase of nth millisecond output
Tcoh
Coherence Time
Tms
Integration time in ms
vm
Maximum Doppler velocity
vs
Doppler velocity of satellite
w
Signal level
xQ,o
Imaginary part of correlation out-of-peak
xI,o
Real part of correlation out-of-peak
xI,p
Real part of correlation peak
xQ,p
Imaginary part of correlation peak
yI,p
Real part of correlation peak after Nc ms
yQ,p
Imaginary part of correlation peak after Nc ms
yI,o
Real part of correlation out-of-peak after Nc ms
yQ,o
Imaginary part of correlation out-of-peak after Nc ms
zo
Correlation out-of-peak after Nnc ms
zp
Correlation peak after Nnc ms
χ2 ()
Chi-square distribution
Chapter 1
Introduction
This chapter gives an introduction to the thesis, beginning with the motivation to investigate the particular research topic. This is followed by a discussion of the research objectives and, finally, the thesis outline.
1.1
Motivation for the research topic
The satellite-based navigation was started in the early 1970s. After some small-scale
system studies, the GPS program was approved in December 1973. Since its launch,
GPS has emerged as the most dominant technology for providing precise location and
navigation capability to the end users. But GPS cannot provide adequate accuracy in
some environments, such as indoor and densely populated urban areas. As a result, there
is a clear requirement for developing a new navigation system that will overcome the
limitations of the GPS and will be compatible with the GPS. The European navigation
system, Galileo is planned to meet the overall requirements. Galileo is expected to operate
by 2010 [15]. Galileo is an initiative of the European Commission and the European
Space Agency (ESA). The new satellites are not yet in the orbit but the signal properties
are already standardized in a first phase so that we can start to analyze the characteristics
of these signals.
Most of the applications of GPS are considered as outdoor applications but nowadays,
the indoor personal navigation applications are getting popularity. In such situations, the
typical GPS receivers suffer degraded performance or sometimes even complete failure
because the signal experiences severe attenuation in the indoor environments. One of
the most important personal positioning applications is the emergency call positioning in
the cellular network, imposed by the authorities. The accuracy is very critical for such
applications. Galileo is planned to increase the accuracy level for such applications. Still
1
Introduction
the characteristics of the indoor propagation need to be well understood to be able to
develop the solution for the indoor positioning problem.
In the outdoors, there are combinations of Line-Of-Sight (LOS) and Non Line-OfSight (NLOS) signals available, whereas in the indoors there is NLOS propagation only.
Most of the time, there is no LOS signals available in the indoors due to the various
obstructions. For the purpose of deriving indoor navigation algorithms, the satellite-toindoor propagation and its fading statistics have great importance. There are several
studies that attempted to develop channel models for GPS-indoor channel. In [35] the
authors used a strong reference outdoor signal to augment indoor processing capabilities
and conduct coherent integrations of up to 160 ms. The existence of deep fades and their
impact on indoor signals were observed. In [48] the author analyzed high bandwidth
raw GPS data with high sensitivity techniques to characterize fading and the multi-path
indoor characteristics. In [27], the authors showed that the GPS-indoor channel fading
amplitudes of the first arriving peak matches well with the Nakagami-m fading model.
In [26, 28], satellite-to-indoor propagation channel characteristics have been analyzed. It
was shown that that the indoor signal is expected to be very weak and embedded in noise.
Thus, long coherent and non-coherent integrations are required.
Pseudolites (PLs), placed on earth surface, especially indoors, are a relatively new
technology with great potential for a wide range of positioning and navigation applications. They can be used either as augmentation of space-based positioning systems or as
independent systems for indoor positioning and capable of showing better performance
[36, 47]. That is why, the PL-to-indoor propagation and its fading statistics have also
great importance.
The receivers should have the capability to estimate the Carrier to Noise Ratio (CNR)
as accurately as possible. In the indoors the signal power remains very low, which affects
the delay estimation accuracy, and, thus, the position accuracy. As a result, the conventional GPS receivers are not able to estimate the CNR accurately for the location services
in moderate indoor reception. Although the topic of CNR estimation in the GPS receivers
is addressed in the literature [22, 33, 38], not much of the published analysis is based
on the correlation of the incoming signal. Also few moment-based CNR estimators are
found in literature, but these estimators have not been developed based on the correlation
function of the BPSK/BOC modulated signals [40]. Furthermore, the author is not aware
of extensive comparisons between different CNR estimation methods.
2
Introduction
1.2
Objective of the thesis
As a part of the Advanced Techniques for Personal Navigation (ATENA) and Future
GNSS Applications and Techniques (FUGAT) projects, the objective of this thesis has
been to analyze the different measurement data captured from different satellites and
pseudolites in different indoor scenarios. The purpose has been to estimate the CNR
accurately and derive a suitable channel model. The ATENA and FUGAT projects are
research projects carried out at Tampere University of Technology (TUT) in cooperation
with some industrial partners during the years 2005-2008. The overall objectives of the
ATENA and FUGAT projects are very wide scale and thus this thesis only covers a very
small part of that.
1.3
Thesis contributions
The novel contributions of the thesis are given below:
• A study on different navigation data bit estimation methods for GPS signals.
• Derivation of three moment-based CNR estimators based on the correlation of the
incoming signal. A comparison of these estimators has been performed including
four other estimators derived in similar way. A procedure has also been proposed
for choosing the required noise samples for estimating the CNR accurately.
• Combined fading channel models have been proposed for matching with the measurement data.
1.4
Thesis outline
This thesis consists eight chapters. The structure of the thesis is given below.
Chapter 1 has introduced the motivation, related previous studies and the overall
objective of the research.
Chapter 2 discusses the currently available fading channel modeling techniques which
are used in the communication systems.
Chapter 3 introduces the GPS and Galileo systems to the readers from the point of
view of signal characteristics.
Chapter 4 discusses the measurement setups for the different indoor measurement
campaigns for GPS based pseudolites and satellites.
3
Introduction
Chapter 5 is dedicated to the measurement data analysis. In the navigation data
estimation part of the thesis, different methods are studied.
Chapter 6 describes a signal model and the derived moment-based CNR estimators
based on the correlation of the incoming signal. The performance of each estimator is
tested for simulation based BOC modulated signal and measurement based BPSK modulated signal. The results for different estimators are compared in order to find the most
robust estimator. A comparative analysis of the navigation data estimation methods described in Chapter 5 is presented in this chapter too.
Chapter 7 presents proposed theoretical fading channel models along with the channel model based on the raw data to the readers.
Chapter 8 finally presents the conclusions of the overall research.
4
Chapter 2
Fading is the term used to describe the fluctuations in the envelope of a transmitted radio
signal. Fading is a common phenomenon in wireless communication channels caused
by the superposition of two or more versions of the transmitted signals which arrive at
the receiver at slightly different times. The resultant received signal can vary widely
in amplitude and phase, depending on various factors such as the relative propagation
time of the waves and bandwidth of the transmitted signal [3, 16]. This chapter starts
by discussing the multi-path channel parameters. Then it discusses the currently available fading channel modeling techniques commonly used in the communication systems. The most commonly known statistical representations of fading are: Rayleigh [48],
Rice [24, 48], Nakagami-m [27, 49], Log Normal [48], and Loo [31] distributions. Finally, it presents the combined fading channel models by combining two or more models.
2.1
Multi-path channel parameters
2.1.1
Time dispersion parameters
In order to compare different multi-path channels, time dispersion parameters such as the
Mean excess delay, τ and Root Mean Square (RMS) delay spread, στ are used. The
mean excess delay is the first moment of the power delay profile and it can be given by
[39]:
∑ P(τk )τk
τ =
k
∑ P(τk )
k
where P(τk ) is the relative amplitude of the multi-path component at kth delay (τ).
5
(2.1)
6
The RMS delay spread is the square root of the second central moment of the power
delay profile and is given by [39]:
q
τ2 − (τ)2
στ =
(2.2)
where
∑ P(τk )τ2k
τ2 =
k
∑ P(τk )
(2.3)
k
2.1.2
Coherence bandwidth
Coherence bandwidth, Bcoh is the maximum transmission bandwidth over which the
channel can be assumed to be approximately constant in frequency. That is, a signal
having frequencies within a bandwidth Bcoh will be affected approximately similarly by
the channel. The RMS delay spread and coherence bandwidth are inversely proportional
to each other. If the coherence bandwidth is defined as the bandwidth over which the
frequency correlation function is above 0.9, then the coherence bandwidth is given by
[29]:
Bcoh ≈
1
50στ
(2.4)
If the frequency correlation function is above 0.5, then the coherence bandwidth is
given by [39]:
Bcoh ≈
1
5στ
(2.5)
2.1.3
7
Doppler shift and Doppler spread
The movement of the satellites introduces frequency shifts on the carrier and the code of
the received signal. This phenomenon is known as the Doppler effect. The maximum
Doppler shift, fD can be given by [45]:
vs f c
(2.6)
c
where vs is the speed of the satellite, fc is the carrier frequency and c is the speed of
light.
The Doppler spread, fds is defined as the range of frequencies over which the received Doppler spectrum is essentially non-zero. An example Doppler spectrum is given
in Figure 2.1 based on Jake’s model. The maximum Doppler shift used in this figure is 10
Hz.
fD =
Doppler power spectrum
1
Normalized Doppler power spectral density
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−10
−8
−6
−4
−2
0
2
4
Frequency shift from the carrier [Hz]
6
8
10
Figure 2.1: Example of Doppler spectrum
2.1.4
Coherence time
Coherence time, Tcoh is the maximum difference in time such that two states of the channel, measured less than Tcoh seconds apart, are still correlated at some extent. The Doppler
spread and coherence time are inversely proportional to each other. A popular rule of
thumb for digital communications is to define the coherence time by [39]:
Tcoh =
where fD is the maximum Doppler shift.
0.423
fD
(2.7)
2.2
8
Fading channel models
The general term, fading is used to describe the fluctuations in the envelope of a received
radio signal. However, when speaking of such fluctuations, it is of interest to consider
whether the observation has been made over short distances or long distances. For a
wireless channel, the former case will show rapid fluctuations in the signal’s envelope,
while the latter will give a more slowly varying, averaged view. For this reason, the first
scenario is formally called small-scale or multipath fading, while the second scenario
is referred to as large-scale fading [5]. Small-scale fading is explained by the fact that
the instantaneous received signal strength is a sum of many contributions coming from
different directions due to many reflections of the transmitted signal reaching the receiver
[3]. Large-scale fading is due to shadowing. Rayleigh and Rician models are the common
small-scale fading models. The Nakagami distribution also falls in this class. Log-normal
can be used for large-scale fading. Loo model combines both small-scale fading and
large-scale fading. A brief overview of the fading channel models is presented in the
following sub-sections.
2.2.1
Rician fading channel
The Rician distribution models the channel in the situation when there is strong LOS
signal with the presence of some weaker, randomly-distributed multipath components.
The envelope of a signal undergoing Rician fading can be expressed by [24, 37]:
¶ µ
¶
(w2 + µ2Rice )
wµRice
fRice (w) = 2 exp −
I0
σRice
2σ2Rice
σ2Rice
w
µ
(2.8)
where fRice (w) is the probability of the signal amplitude level w, σ2Rice is the variance (can
be given by σ2Rice = (var(I) + var(Q))/2, where I and Q are the in-phase and quadrature
p
components of LOS coefficient), µRice = mean(I)2 + mean(Q)2 , and I0 (x) is the modified 0th order Bessel function and can be given by [9]:
µ ¶2m
ix
(−1)m
I0 (x) = ∑
m=0 m!(m + 1) 2
∞
(2.9)
where i is the imaginary unit.
The Rician factor KRice (i.e., the ratio of LOS to multi-path power) is given by KRice =
µ2Rice
[37]. Example of Rician distributions is given in Figure 2.2.
2σ2
Rice
9
Rician PDF with σ
Rice
=1
0.7
µ
Rice
=0
µRice=1
µRice=2
0.6
µ
Rice
=4
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
Figure 2.2: Example of Rician distributions for different µRice
2.2.2
Rayleigh fading channel
Rayleigh represents the worst case fading case and it can be considered as special case
of Rician distribution when no LOS component is present. The envelope of a signal
undergoing Rayleigh fading can be expressed by [37, 48]:
fRayl (w) =
w
σ2Rayl
µ
w2
exp − 2
2σRayl
¶
(2.10)
q
p
where σ2Rayl is given by σ2Rayl = π2 mean( I 2 + Q2 ) [37]. An example of Rayleigh
distributions is given in Figure 2.3.
Rayleigh PDF
1.4
σ
=0.5
σ
=1.0
Rayl
Rayl
σRayl=1.5
1.2
σ
=2.0
Rayl
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
Figure 2.3: Example of Rayleigh distributions for different σRayl
2.2.3
10
Log-normal fading channel
Signals that propagate through some attenuating medium have a log-normal power distribution [13]. If z is the signal amplitude, the lognormal distribution can be expressed by
[48]:
¶
µ
(log10 (w) − µLogn )2
1
fLogn (w) = √
exp −
2σ2Logn
w 2πσLogn
(2.11)
where σLogn represents the standard deviation of log10 (A) and µLogn represents the mean
p
of log10 (A), where A = I 2 + Q2 is the envelope corresponding to LOS delay. The parameters σLogn and µLogn depend on the medium of propagation and possibly on the motion of transmitter and receiver. An example of Log-normal distributions with various
σlogn is given in Figure 2.4.
Log−normal PDF with µlogn = 1
0.7
σlogn=0.5
σlogn=1
σlogn=1.5
0.6
σ
=2.0
logn
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
Figure 2.4: Example of Log-normal distributions for different σlogn
2.2.4
Loo fading channel
Loo [31] has developed a statistical distribution assuming that the fading of the shadowed
LOS signal is log-normally distributed and the multi-path signals’ fading is Rayleigh
distributed. Loo’s distribution model can be expressed by [24, 48]:
ÃZ
µ
(log10 (x) − µLogn )2
1
fLoo (w) =
exp −
2σ2Logn
0 x
!
¶ ³
µ
¶
´
(x2 +w2 )
xz
w
√
− 2σ2
I0 σ2 dx
σ2
2πσ
∞
Loo
Loo
Loo
Logn
(2.12)
11
where σLogn = std(log10 (A)) and µLogn = mean(log10 (A)) represent the standard deviation and mean of the logarithm of the measured envelope A, respectively, and σLoo =
std(A), µLoo = mean(A) are the standard deviation and mean of the measured envelope,
respectively. An example of Loo distributions with various σLoo is given in Figure 2.5.
Loo PDF
0.25
σLoo=0.5
σLoo=1
σLoo=1.5
0.2
σLoo=2.0
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
Figure 2.5: Example of Loo distributions for different σLoo
2.2.5
Nakagami fading channel
Three types of Nakagami distributions are found in the literature namely Nakagami-q,
Nakagami-n and Nakagami-m distributions. The Nakagami-q distribution is also known
as Hoyt distribution. It can span from one-sided Gaussian fading (q=0) to Rayleigh fading
(q=1). The Nakagami-n distribution is also known as the Rician distribution. The Rician
factor of the Rician distribution and the parameter, n of the Nakagami-n distribution are
related by KRice = n2 . The Nakagami-n distribution spans the range from Rayleigh fading
(n = 0) to no fading (n = ∞) [42]. Nakagami-m distribution is a generic model of fade
statistics that is used in the study of mobile radio communications [7, 32]. A wide class
of fading channel conditions can be modeled with Nakagami-m distribution [32]. This
fading distribution has gained a lot of attention lately, since the Nakagami-m distribution
often gives the best fit to land-mobile [2, 41, 43] and indoor mobile multi-path propagation
as well as scintillating ionospheric radio links [42].
The PDF of a Nakagami-m fading amplitude can be expressed by [25, 49]:
µ
¶m
µ
¶
m
mw2
2
2m−1
w
exp −
,
fNaka (w) =
Γ(w) µNaka
µNaka
(2.13)
12
where µNaka = mean(|α|2 ) = mean(A2 ) is the mean of the envelope power (α is the complex channel coefficient), m is the Nakagami-m factor and Γ(.) is the Gamma function.
The following estimate of m factor can be used (i.e., m is equal to the inverse of the
normalized variance of the squared envelope) [25, 37]:
m=
µ2Naka
mean(A2 − mean(A2 ))2
(2.14)
For m = 1, Nakagami-m is equivalent to Rayleigh distribution [32]. Nakagami-m distribution can closely approximate the Nakagami-q distribution by a one-to-one mapping
between m parameter and the q parameter. The mapping is given by [42]:
m=
(1 + q2 )2
2(1 + 2q4 )2
(2.15)
Another one-to-one mapping can be found between the m parameter and the n parameter allowing the Nakagami-m distribution to closely approximate the Nakagami-n
distribution. The mapping is given by [42]:
m=
(1 + n2 )2
1 + 2n2
(2.16)
An example of Nakagami-m distribution with various m and µNaka values is given in
Figure 2.6.
Nakagami PDF
1.4
m=1,µ
=1
Naka
m=1.0,µ
=2
Naka
m=2,µNaka=3
1.2
m=3,µNaka=1
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
Figure 2.6: Example Nakagami distribution for different m and µNaka values
Chapter 3
This chapter presents the concepts of GPS and Galileo systems. It first starts with the
basic position measurement concepts that have been used in the GPS. Then, it describes
the modernized GPS system, and finally, the new European navigation system, Galileo
that is planned to be launched within few years.
3.1
Basic GPS overview
The GPS system is based on the time-of-arrival measurements. The distance between a
satellite and a receiver is calculated from the knowledge of how much time it takes for
the signal from the satellite to get to the receiver [8]. The signal from the satellite to
the receiver travels at the speed of light. If we know the travel time of the signal from
the satellite to the receiver, we can easily calculate the distance. Figure 3.1 demonstrates
the distance based positioning in two-dimensional case. In order to determine the receiver
position, three distances from three satellites are required. Two satellites give two possible
solutions because two circles intersect at two points. Hence, a third satellite is needed to
determine the receiver position uniquely. For similar reason, to calculate the position in
the three-dimensional plane, four satellites and four distances are required.
In the above discussion, it is assumed that the distance measured from the satellite to
the receiver is very accurate and does not contain any bias error. But actually the measured
distance has an unknown bias error because the receiver clock and the GPS clock are not
fully synchronized. In order to resolve this bias error, one more satellite is required and
thus in order to find the accurate position five satellites are needed [45]. However, the
general statement is that four satellites can be used to find the receiver position, even
though the measured distance has a bias error [45].
Although, it is enough to know four distances in the process of getting an accurate
13
Satellite 1
14
Satellite 2
Receiver
Satellite 3
Figure 3.1: Two-dimensional user position.
positioning result, it is not enough to have four random satellites flying in the sky. The
satellites must also know their positions. A global network of ground stations is needed
to give the position data to the satellites. In general, the GPS system can be considered as
comprising three segments: the space segment, the control segment and the user segment.
The space segment contains all the satellites. The basic GPS system has a total of 24
satellites divided into six orbits. Each orbit has four satellites and has an inclination angle
of 55-degree. The orbits are separated by 60 degrees and each orbit has a radius of 26,560
km. The control segment consists of five control stations. The purpose of this segment is
to monitor the performance of the GPS satellites, generate and upload the navigation data
to the satellites. The user segment consists of the GPS receivers and the user community.
There are two main signals: the coarse/acquisition (C/A) and the precision (P) codes.
The P code is modified by Y code, which is refereed as P(Y) code. P(Y) is used for
military purpose and it is not available for civilian users. The basic GPS signal contains
two carrier frequencies: L1 and L2. The center frequency of L1 is at 1575.42 MHz and L2
is at 1227.6 MHz. L1 frequency contains C/A and P(Y) signals and L2 frequency contains
only the P(Y) signal. This thesis focuses mostly on the civilian C/A signal. The C/A code
is BPSK modulated with a chip rate of 1.023 MHz while the chip rate of P(Y) signal
is 10.23 MHz. The navigation message, which contains information about the satellites,
GPS time, clock behavior and system status is modulated on both the L1 and L2 carriers
at a chip rate of 50 bits per second (bps) with a bit duration of 20 ms.
3.2
Modernized GPS system overview
The only navigation system that can be used worldwide is the GPS system. GPS is military operated, but it is used for many commercial and civilian services nowadays. It
shows very poor performance for the indoor location based services too. As a result GPS
does not offer enough accuracy or warranty of service and it cannot be used in many vital
positioning applications. The GPS system has been upgraded to meet the requirements,
but still its functionality cannot be trusted in many scenarios. The modernized GPS frequency plan is shown in Figure 3.2. The modernized GPS includes a new frequency band
L5 (1176.45 MHz) that provides a wide-band signal. In addition, the new L2C signal
will provide the civilian community a more robust signal that is capable of improving
resistance to interference and allowing longer integration times to the receivers. A new
military M-code will also be added to L1 and L2 bands, but will be spectrally separated
from the civil codes. It has been decided that the new modulation type for the new M
signal will be Binary Offset Carrier (BOC) modulation.
Figure 3.2: The modernized GPS frequency plan [14].
15
3.3
Galileo system overview
The European navigation system, Galileo is planned to achieve European sovereignty and
service guarantees through dedicated system under civil control [15]. The overall Galileo
system consists of 30 satellites (27 operational+3 active spares), positioned in three circular Medium Earth Orbit (MEO) planes at 23,222 km altitude above the Earth, and at an
inclination angle of the orbital planes of 56 degrees [5]. The services that will be provided
by the Galileo are: Open Service (OS), Safety of Life Service (SoL), Commercial Service (CS), Public Regulated Service (PRS) and Search and Rescue Service (SAR). The
reliability of the Galileo services is higher than that of the GPS [44]. Galileo is meant to
provide better navigation accuracy due to its signal properties. The BOC modulation is
planned to be used in the Galileo signals [5].
The frequency plan for the Galileo system is shown in the Figure 3.3 which consists of
four frequency bands: E5a, E5b, E6 and E2-L1-E1. The E2-L1-E1 band with the center
frequency 1575.42 MHz is the most interesting band as the current GPS signal (C/A) is
in it. This thesis mostly focuses on this frequency band. The readers who are interested
in other frequency bands are referred to [17]. Both the GPS C/A code and Galileo OS
signals are transmitted in the same frequency band. But still the signals do not interfere
significantly with each other since different modulation is used. The most important characteristics of the Galileo signals, in comparison with the GPS signals, are the different
modulation types and code lengths. SinBOC(1,1) (briefly denoted as SinBOC) has been
the candidate modulation type for the Galileo OS signal in the E2-L1-E1 band for many
years. The code length for the OS signal is 4092 chips, which is four times higher than
the GPS C/A code length. Recently the GPS-Galileo working group on interpretability and compatibility has recommended an optimized Multiplexed Binary Offset Carrier
(MBOC) spreading modulation that would be used by Galileo for its OS service and also
by GPS for its L1C signal [18]. However, this thesis presents the simulation results for
the SinBOC(1,1) only. For the technical details of the BOC modulation, the readers are
referred to [18],[4] and [5].
16
Figure 3.3: Galileo frequency plan [46].
The use of GPS and Galileo at the same time is very interesting. The accuracy can
be increased a lot by using the two systems together. The indoor reception might be
improved in this way to provide the location based services to the users.
17
Chapter 4
This chapter presents the measurement campaign descriptions to the readers. It first starts
with the pseudolite (PL) based measurement campaign where single PL-based and multiple PL-based measurement campaign descriptions are discussed. Then it discusses the
satellite-based measurement campaign. These measurements were captured with the help
of Space Systems Finland (SSF) and u-Nav Microelectronics.
4.1
Pseudolite based measurement campaign
Using PLs, two types of measurement campaigns were undertaken: single PL-based and
multiple PL-based. These measurement setup descriptions are given in the following
subsections. In these measurements, PRN indexes higher or equal to 32 were used, which
are mostly reserved for non-satellite use. The sampling frequency of the GPS receiver was
16.36 MHz. The L1 carrier in the data was down-converted to an intermediate frequency
(IF) of 41552 Hz.
4.1.1
Single pseudolite based measurement
The measurements were first carried out in the Tamppi arena building, then in the Festia
building of TUT, Finland during June, 2005. Two synchronized GPS receivers were used.
One receiver was used as reference receiver and was connected to the PL via cable. The
other receiver was connected to an indoor antenna measuring the signal coming from the
air. The transmit antenna was placed in a fixed position at the first floor, with an elevation
of around 7 meters with respect to the receiver. The radiation patterns of the helix antenna (transmit antenna) used in the measurements was Right Hand Circular Polarization
(RHCP), where the main beam was within ±30/35 from the antenna pointing direction.
18
Attenuation in PL software was 20 dB.
The measurement process was carried out in 5 sets:
• SET − 1: It was captured in the Tamppi arena sports hall. The receiver was moved
from inside the main beam to the outside. The photo of the environment is shown
in Figure 4.1 and the schematic representation of the measurement set is shown in
Figure 4.2.
Figure 4.1: Photo taken in the Tamppi arena sports Hall from the transmitter position.
• SET − 2: It was also captured in the Tamppi Arena sports hall. But this time the
receiver was moved inside the main beam only. The schematic representation of the
measurement set is shown in Figure 4.2.
• SET − 3: It was the last set that was captured in the Tamppi Arena sports hall.
The receiver was moved outside of the main beam. The antenna pointing direction
was parallel to the direction of the movement. The schematic representation of the
measurement set is shown in Figure 4.3.
• SET − 4: It was captured in the Festia main hall. The receiver movement was
within the main beam with few times out of LOS.
19
20
Antenna Pointing Direction
35
35
7m
7m
Receiver
Receiver
Receiver
9m
6m
9m
6m
Figure 4.2: Transmitter and receiver positions for the PL-based indoor propagation measurement in the Tamppi Arena. Left: during SET-1, SinglePL. Right: during SET-2,
SinglePL.
Sports Hall
Receiver
Figure 4.3: Transmitter and receiver positions for the PL-based indoor propagation measurement in the Tamppi Arena during SET-3, SinglePL.
Figure 4.4: Transmitter and receivers position for the PL-based indoor propagation measurement in the Fiesta building during SET-4, SinglePL and SET-5, SinglePL.
• SET − 5. This set was also captured in Festia main hall. The receiver movement
was within NLOS condition (almost always behind obstructions).
In Figure 4.4, two pictures of the environment taken both from the transmitter antenna position and receiver antenna position are shown for SET-4, SinglePL and
SET-5, SinglePL. The schematic representation of these measurement sets is shown
in Figure 4.5.
4.1.2
Multiple pseudolite based measurement
Multiple PLs based measurements were also carried out in TUT during November, 2005.
The measurements were first carried out in Tietotalo main corridor, and then in the Institute of Communications Engineering (ICE) offices in 5 sets:
• SET − 1: It was captured in Tietotalo main corridor with 2 active PLs: PL1 (PRN
33, used as reference) and PL3 (PRN 34) were placed according to Figure 4.6. The
PLs were placed in the second floor at a height of 5 meters, and the receivers were
in the ground level. The receiver movement was started from 10.5 m away from
PL1, first towards PL3, then towards PL1, and so on. The attenuation in PL1 and
PL3 were about 55- 60 dB.
21
Obstructions
22
Obstructions
1st
Festia Main Hall
Level
Receiver
Ground Level
Figure 4.5: Transmitter and receivers position for the PL-based indoor propagation measurement in the Fiesta Main Hall during SET-4, SinglePL and SET-5, SinglePL.
• SET − 2: This set was captured in the same environment setup as SETmultiple − 1.
But one more active PL was used: PL2 (PRN 32, attenuation 50 dB). The receiver
was first moved towards PL1, then towards PL3 and at last below the bridge where
PL2 was placed.
• SET − 3: This set was also captured with three active PLs (PL1: PRN 33, attenuation 55 dB; PL2: PRN 32, attenuation 40 dB, PL3: PRN 34, attenuation 55 dB).
PL1 and PL3 were placed at the same location as before, but PL2 was placed one
level up, at the third floor, above the window ceiling of Tietotalo.
• SET − 4: It was captured in the office corridor of ICE with three active PLs. They
were placed in a triangle. PL1 also used as reference (PRN 33, attenuation 60 dB).
PL2 and PL3 attenuation was 60 dB and 55 dB respectively. The receiver movement
was along the corridor.
• SET − 5: It was captured with the same configuration as for SET − 4, MultiplePL
(same attenuation and same PRNs), but PL2 was used as reference.
Figure 4.7 shows a schematic representation of the measurement setup for SET − 4,
MultiplePL and SET − 5, MultiplePL.
23
4m
15.8 m
15.8 m
PL1
PL3
H=5 m
H=5m
Receiver
PL2
H=5m
Figure 4.6: Schematic representation of measurement SET − 1, SET − 2 and SET − 3,
MultiplePL (the PLs are shown in red and the receiver in black).
PL 3
PL 2
H=1.4m
6.1 m
H=0.97m
Offices
Offices
15.4m
PL 1
H=1.45m
Receiver
Offices
Figure 4.7: Schematic representation of measurement SET − 4, MultiplePL and SET − 5,
MultiplePL.
4.2
Satellite based measurement campaign
Two satellite based measurement campaigns were undertaken by TUT and u-Nav microelectronics, Finland during March, 2004. In both campaigns, the transmitters were the
different GPS satellites available in view during the measurement date and the receivers
were the integrated GPS receivers with sampling rate of 16.36 MHz. Two GPS receivers
synchronized to a common clock, operating in parallel were used. The first one was
used to acquire the signal from an outdoor antenna placed on the roof of the building.
This signal was quite strong, and it was used as the reference signal for code-phase and
Doppler frequency acquisition, as well as for frequency drift estimation and correction.
The second receiver was moved in the indoor environment to capture the indoor signal.
The down-converted intermediate frequency (IF) was same as the PLs.
Among the two measurements, one was carried out in typical office-room scenario and
the other was carried out in typical office-corridor scenario. The first scenario, denoted by
RoomSAT , shown in Figure 4.8 (left), corresponds to a small room without any window
(about 5m2 ), where in the front there was small corridor with large windows. Here the
LOS signal is more likely to be absent. The second scenario, denoted by CorridorSAT ,
shown in Figure 4.8 (right), corresponds to a long corridor with open windows and doors.
The receiver movement inside the environments was random and it was at the walking
speed. All the measurements were taken for a duration of 1-2 mins for reliable statistics.
Figure 4.8: Photo of RoomSAT (left) and CorridorSAT (right) scenario.
24
Chapter 5
This chapter discusses the data analysis steps that are followed to compute the Channel
Impulse Response (CIR) estimates. The setup block diagram is detailed in Figure 5.1. An
initial Doppler drift estimate (incorporating the drift due to low IF sampling) and an initial
delay estimate are obtained based on the reference signal by scanning the whole delayDoppler space. Also, the code drift, frequency drift and navigation data are estimated
based on this reference signal and then removed from the wireless signal. Then, the wireless signal is correlated with the replica code, by taking into account the delay-Doppler
estimates and their drifts. In the indoor environment the coherent integration must be
longer than 20 ms in order to compensate for the increase in the noise level. For this
reason, the removal of the navigation data must be done before the coherent integration,
similar with [26], [27] and [28]. The most important parts of Figure 5.1 are discussed in
the following sections.
Indoor
signal
replica GPS code
Ref
Correlators
signal bank
Doppler, delay
and drift
estimates
Correlation
on 1 ms
Data
removal
Integration
on Nc ms
CIR estimates
Navigation
data
estimates
Figure 5.1: Block diagram of the measurement data processing.
25
5.1
26
Acquisition of C/A code
For any DS-SS (Direct sequence Spread Spectrum) system, it is necessary to estimate
the timing and the frequency shift of the received signal in order to be able to de-spread
the received signal and to obtain the original data. For that, it is required to define the
position where there is an alignment between the received signal and the spreading code
[23]. This is done to estimate the Doppler shift and initial delay. The process is done
using cross-correlation, which measures the similarity of the code and the delayed replica
of the same code. The search process is usually two-dimensional by which both the time
shift (i.e., delay) and Doppler shift can be determined [22]. The value of the Doppler shift
changes over time according to the place and speed of the satellite. It is naturally much
more easier to look for the correct frequency, if the probable Doppler shift is known in
advance. According to [45] the maximum Doppler velocity, vs of the satellite is 929 m/s.
The Doppler frequency caused by the land vehicle is often very small. For a stationary
observer the maximum Doppler shift on the carrier is
vs f c
c
929 × 1575.42 × 106 ∼
=
= 5kHz
3 × 108
fD =
where c is the speed of light and fc is the L1 carrier frequency. In the measurement
campaigns, the receiver was moved in a low speed which was around 1 − 2 km/hr. So the
maximum Doppler shift was around ±5 kHz. However, if a receiver is moved by using
a high speed aircraft, it is reasonable to assume the maximum Doppler shift is ±10 kHz
[45].
On the other hand the Doppler shift on the C/A code is quite small because of the
low frequency of the C/A code. The C/A code has a frequency of 1.023 MHz which is
1, 540 (1575.42/1.023) times lower than the carrier frequency. For a stationary observer
the maximum Doppler shift on the C/A code is
vs fc
c
929 × 1.023 × 106 ∼
=
= 3.2Hz
3 × 108
fD =
where fc is the C/A code frequency. If the receiver is moved at high speed, the Doppler
shift can be assumed as ±6.4 Hz.
27
In the search process, all possible code delays and frequencies are searched through
with some predefined search steps. The search space is typically equal to the length
of the spreading code in the code-delay domain. In the Doppler frequency domain, the
search interval can be several kHz or even tens of kHz [11]. If some a priori information
(i.e., assistance data) about the Doppler frequency is available, the frequency interval
may be just few tens of Hz [11]. In order to accomplish the search in a short time, the
bandwidth of the searching program cannot be very narrow. Using a narrow bandwidth
for searching means taking many steps to cover the desired frequency range and it is time
consuming. On the other hand, searching with wide bandwidth provides low sensitivity.
So the bandwidth should be selected properly. For measurement data, 1023 ∗ 16 samples
are searched in the initial stage with a frequency step of 400 Hz. In the next stage a
window of 200 correlators is used with smaller frequency step of 20 Hz to get better
phase estimate and check the correctness of the frequency estimated from the previous
stage. The correct delay of the C/A code for SET-1, SinglePL is shown as an example in
Figure 5.2 after the initial stage.
1
0.9
Normalized correlation power
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−600
−400
−200
0
Delay [chips]
200
400
600
Figure 5.2: The correct delay of C/A code of SET-1, SinglePL.
5.1.1
Search window
Each tentative code-phase is called a code bin (or a time bin) and each tentative frequency
shift is denoted as a Doppler bin (or a frequency bin). One code bin together with one
Doppler bin compose a search bin (or a test cell). The whole code-frequency uncertainty
region can be divided into several search windows and each window can be divided into
several time-frequency bins. The time-frequency search window defines the decision re-
gion [22].
5.1.2
Search strategy
In the search stage, the search windows are examined to see whether the time-frequency
estimate is correct or not. The search process is started from one search window, with a
certain tentative Doppler frequency and a certain tentative delay. All delays and frequencies, which correspond to the size of the search window at issue, are searched through
with the predefined search steps. If the window is decided to be dismissed, the search
process moves on to the next search window, and the same procedure is continued, until
the correct window and the correct delay-frequency combination is found [22, 23]. A
hybrid search is used in this thesis.
5.1.3
Correlation
The tentative time-frequency bins are tested and possible signals are detected via crosscorrelation. This means that the received signal is correlated with the reference code with
different tentative delays and frequencies, and the resulting values are then combined to
achieve a two-dimensional correlation output for the whole search window. From the
correlation output, it can be further determined whether the search window is correct
or not via a correlation peak which appears for correct delay-frequency combination.
The correlation process is described in detail in [22, 34]. The correlation properties of
the spreading codes are very important. If the auto and cross-correlation properties are
perfect, the correlation function would appear as a pure impulse at the correct delay and
will have zero values elsewhere. But actually there is always some interference and noise
present, which affects the correlation output of the received signal and reference code. An
example correlation function is shown in Figure 5.3 for SET-1, SinglePL data.
The initially estimated carrier and delay for each set of PL signals are shown in Table
5.1. Also the carrier and delay for each set of satellite signal for different PRNs are shown
in Table 5.2.
5.2
Drift estimation
As discussed in the previous section, the satellite orbital motion can cause a Doppler shift
up to 5 kHz for stationary receiver. In addition, satellite clock drift affects the actual
frequency emitted from the GPS satellites, causing a further Doppler effect. The carrier
phase estimation is highly affected by the frequency and phase drifts. According to [48],
28
29
1
0.9
Normalized correlation power
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−8
−6
−4
−2
0
Delay [chips]
2
4
6
8
Figure 5.3: One dimensional (left) and two dimensional (right) correlation of C/A code
for SET-1, SinglePL after 1ms integration.
Table 5.1: Delay and Carrier for different sets of PL signal.
Set
1
2
3
4
5
SinglePL
Delay (ms) Carrier (Hz)
263.95
39200
560.07
39190
604.22
39290
242.33
39270
311.99
39300
MultiplePL
689.62
39210
611.07
38990
645.33
39440
621.70
39420
646.72
39400
Table 5.2: Delay and Carrier for satellite data.
PRN
3
15
16
18
21
26
CorridorSAT
980.58
42620
911.63
42800
752.64
38600
197.76
36400
621.39
39400
311.39
42420
RoomSAT
511.38
43200
620.21
40020
711.11
39800
211.56
38200
525.24
39420
325.33
42100
30
a 2nd order polynomial fitting model can be used to estimate the drifts. In the drift estimation of the measurement data, 2nd order polynomial fitting model is used. The estimated
phase and the effects of multi-paths on phase estimation for a few data sets are discussed
in Appendix A.
5.3
Navigation data bit estimation
The C/A code is a bi-phased code signal which changes the carrier phase between 0 to π
at a rate of 1.023 MHz. The navigation data bit is also bi-phased code but its rate is 50
Hz. Each data bit is 20 ms long. Since C/A code is 1 ms, there are 20 C/A code symbols
in one data bit. Thus, in one data bit all 20 C/A codes have the same phase. It is necessary
to find the phase transitions to estimate the navigation bits. For this purpose the following
two methods are compared:
• Threshold approach
• Signum (sgn) function approach
The phase difference between the adjacent millisecond outputs can be represented by:
∆θ = θn+1 − θn
(5.1)
In the threshold approach three thresholds (π/2, π, 2π/3) are studied. The idea is that
if the absolute phase difference between adjacent millisecond outputs is beyond a certain
threshold, there is a data transition. These points are the navigation data points. After
finding the navigation points, the navigation data are estimated and designated as +1 and
-1. The sign of first navigation bit is arbitrarily chosen. The sub-sequent bits are chosen
based on the thresholds. For example, if abs(∆θ) ≤ π/2, the current navigation bit has the
same sign as the previous bit, otherwise the current bit has different sign. In the signum
function approach, the sign of the phase difference is directly mapped to the navigation
data bits (i.e., +1 and -1).
For each method, the CIR envelope is computed and CNR is estimated in order to find
the most accurate method based on the estimated CNR variance in the overall data. The
results are discussed in Section 6.6 of Chapter 6. The following steps are used to convert
the phase transition to navigation data:
1. Find all the navigation data transitions. The beginning of the first navigation data
should be within the first 20 ms of output data because the navigation data are 20 ms
31
long. The first phase transition is used to find the beginning of the first navigation data.
The first phase transition detected in the output data is the beginning of the navigation
data. If the first phase transition is within the first 20 ms of data, this point is also the
beginning of the first navigation data. If the first phase transition occurs at a later time,
a multiple number of 20 ms is subtracted from it. The remainder is the beginning of the
first navigation data. For simplicity let us just call it the first navigation data point instead
of the beginning of the first navigation data. The first navigation data point can be padded
with data points of the same sign to make the first navigation data point always occur
at 21 ms. This approach creates one navigation data point at the beginning of the data
from partially obtained information. For example, if the first phase transition occurs at
97 ms, by subtracting 80 ms from this value, the first navigation data point occurs at 17
ms. These 17 ms of data are padded with 4 ms of data of the same sign to make the
first navigation data 20 ms long. This process makes the first navigation data point at 21
ms. This operation also changes the rest of the beginnings of the navigation data by 4
ms. Thus, the navigation data points occur at 21, 41, 61, and so on. Figure 5.4 illustrates
the above example. The adjusted first navigation data point at 21 ms is stored. If the first
phase transition occurs at 40 ms, by subtracting 40, the adjusted first navigation data point
occurs at 0 ms. Twenty-one ms of data with either + or - can be added in front of the first
navigation data point to make it occur at 21 ms.
st
st
1 navigation
1
data point at 17
shift occurs at
ms
97 ms
37
57
phase
77
Padded with 4
ms of data
21
41
61
81
101
st
Adjusted 1
navigation data
at 21 ms
Figure 5.4: Adjustment of the first navigation data point.
2. Once the navigation data points are determined, the validity of these transitions
must be checked. These navigation data points are separated by multiples of 20 ms. If
32
these navigation data points do not occur at a multiple of 20 ms, the data contain errors
and should be discarded.
3. After the navigation data points pass the validity check, these outputs are converted
into navigation data. Every 20 outputs (or 20 ms) convert into one navigation data bit. The
sign of the first navigation data is arbitrarily chosen. The navigation data are designated
as +1 and −1.
5.4
Coherent integration
The GPS signals are expected to be rather weak. The common approach to find a weak
signal is to increase the acquisition data length. The advantage of this approach is the
improvement in signal-to-noise ratio. One simple explanation is that an FFT with 2 ms of
data produces a frequency resolution of 500 Hz in comparison with 1 kHz resolution of 1
ms of data. Since the signal is narrow band after the spectrum is de-spreaded, the signal
strength does not reduce by the narrower frequency resolution. Reducing the resolution
bandwidth reduces the noise to half; therefore, the signal-to-noise ratio improves by 3 dB.
For the measurement based data, high coherent integration lengths (50, 200) are used. An
example of CIR snapshots is given in Figure for PRN 3, RoomSAT . The left plot of this
figure shows the CIRs without applying coherent integration and the right plot shows the
CIRs after applying the coherent integration of 200 ms.
Snapshot of CIR envelope, Nc=200ms
Snapshot of CIR envelope, N =1ms
c
1
1
Reference
Indoor
0.9
Reference
Indoor
0.8
0.8
CIR envelope
CIR envelope
0.7
0.6
0.5
0.4
0.6
0.4
0.3
0.2
0.2
0.1
0
−8
−6
−4
−2
0
2
Delay error [chips]
4
6
8
0
−10
−5
0
Delay error [chips]
5
10
Figure 5.5: Example snapshot of correlation based CIR envelope, PRN 3, RoomSAT .
Chapter 6
CNR Estimation
This chapter starts by describing a signal model for BOC/BPSK modulated signals, then
it describes the derivation of the moment based CNR estimators based on the correlation
of the incoming signal. Finally, the performance of each estimator for simulation based
BOC modulated signal and measurement based BPSK modulated signal are presented to
the readers. An initial analysis of such moment based CNR estimators assuming a signal
model is presented in [21]. More detailed analysis including a procedure of choosing
appropriate number of noise samples are presented in this thesis.
6.1
Signal model
As described in the starting part of this chapter, the signal model used here has been
presented in [21]. All the CNR estimators are derived based on this signal model. So, the
signal model has been discussed here again for better understanding.
6.1.1
Coherent integration outputs
Suppose xI and xQ denote the real and imaginary parts of the correlation function after
1 ms integration computed between the incoming BOC/BPSK-modulated signal and the
reference BOC/BPSK-modulated codes. If it is assumed that the channel is Additive
White Gaussian Noise (AWGN) with double sided power-spectral density N0 /2, then the
I and Q components of the noise are Gaussian distributed with 0 mean and variance N0 /2.
The complex noise variance is N0 /2 + N0 /2 = N0 . It is also assumed that the signal power
is Eb , that is the Signal to Noise Ratio (SNR) is Eb /N0 and the Carrier to Noise ratio in 1
33
CNR Estimation
34
kHz bandwidth is [6]:
Eb
+ 10log10 (Bw )
N0
Eb
= 10log10 + 30
N0
CNR[dBHz] = 10log10
(6.1)
where Bw = 1 kHz bandwidth. The statistics of the correlation function after 1 ms integration obey a normal (Gaussian) distribution N (mean, variance): In the correct bins
(peaks) [30]:

¶
µ
√

N0

 xI,p ∼ N F (∆τ, ∆ fD ) Eb , 2
µ
¶
(6.2)

N
0

x
∼
N
0,
 Q,p
2
In the incorrect bins (out-of-peaks) [21]:



 xI,o


 xQ,o
µ
¶
N0
∼ N 0, 2
µ
¶
N0
∼ N 0, 2
(6.3)
Above, F (∆τ, ∆ fD ) can be given by [21]:
F (∆τ, ∆ fD ) = RBOC/BPSK (∆τ)sinc(π∆ fD Tms ),
(6.4)
where sinc(x) = sin(x)/x, ∆τ is the delay error, ∆ fD is residual Doppler error coming
from acquisition stage, RBOC/BPSK (∆τ) is the BOC/BPSK autocorrelation function and
Tms is the 1-ms integration time. For simplicity reason, it is assumed from now on that the
residual Doppler error is 0 and the bit energy (or signal power per bit) in a correct bin is
denoted by:
2
Eeb = RBOC/BPSK
(∆τ)Eb .
(6.5)
There might be several correct bins, according to the steps of scanning the time axis. The
maximum peak corresponds to 0 delay error (RBOC/BPSK (∆τ) = 1 and Eeb = Eb ). The
statistics after 1 ms integration can be written as [21]:

¶
µq

N
0

Eeb , 2
∼ N
 xI,p
µ
¶
(6.6)

N
0

 xQ,p , xI,o , xQ,o ∼ N 0, 2
where the subscript p stands for a peak value and the subscript o stands for an out-ofpeak value.
CNR Estimation
35
If yI and yQ are denoted as the real and imaginary parts of the correlation function
after coherent integration on Nc ms, yI and yQ can be represented [21]:




 yI,k =



 yQ,k =
1
Nc
Nc
∑ xI,i+kNS
i=1
Nc
1
Nc
(6.7)
∑ xQ,i+kNS
i=1
where NS is the oversampling factor. yI,Q are still Gaussian
distributed, of variance
q
N0
1
N N0 = 2N
, and mean 0 (for yQ,p ,yI,o , and yQ,o ) or N1c Nc Eeb = Eeb (for yI,p ). Thus, the
Nc2 c 2
c
statistics after coherent integration can be written as [21]:

µq
¶

N

∼ N
Eeb , 2N0c
 yI,p
µ
¶
(6.8)

N
0

 yQ,p , yI,o , yQ,o ∼ N 0, 2Nc
6.1.2
Non-coherent integration outputs
If the output after non-coherent integration on Nnc blocks is denoted by z, it can be represented by [21]:
1
z =
Nnc
µ Nnc
∑
k=1
y2I,k +
Nnc
∑
k=1
¶
y2Q,k
Nnc
=
∑
k=1
µ
y
√k
Nnc
¶2
(6.9)
where yk is the complex modulus (magnitude) of yI,k + yQ,k . In Equation (6.9), there
is a sum of squares of Gaussians of equal variance σ2 = N1nc var(y) = 2NNc N0 nc . Thus z is
a chi-square distributed variable [37], either central or non-central according to the bin
placement (peak z p or out-of-peak zo ):

µ
¶

N
2
0

 z p ∼ χnc Eeb , 2Nc Nnc , 2Nnc
µ
¶
(6.10)

N
2
0

 zo ∼ χc 2Nc Nnc , 2Nnc
where χ2nc (s2 , σ2 , n) is a non-central chi-square distribution with n = 2Nnc degrees of freee
dom, underlying variance σ2 = 2NNc N0 nc and non-centrality parameter s2 = Nnc NEncb + 0 = Eeb .
And χ2c (σ2 , n) above is a central chi-square distribution with n = 2Nnc degrees of freedom
and underlying variance σ2 = 2NNc N0 nc . According to [37], the mean, second-order moment
and variance can be derived as follows:
CNR Estimation
36
Central distributions:

E(zo )
= nσ2 = NN0c



E(z2o )
= 2nσ4 + n2 σ4 =


 var(z2 ) = 2nσ4 = N02
o
N N2
N02 1
(
Nc2 Nnc
+ 1)
(6.11)
nc c
Non-central distributions:


E(z p )
= nσ2 + s2 = NN0c + Eeb




2
4
2 s2 + (nσ2 + s2 )2

= 2nσ
 E(z p )
µ + 4σ ¶µ
¶
N0
1
1
e
= Nc Nnc + 1 2Eb + Nc + Eeb2





2

 var(z2 ) = 2nσ4 + 4σ2 s2 = N0 2 + 2N0 Eeb
p
Nc Nnc
N N
(6.12)
nc c
Above, E(·) is the expectation operation. For ergodic processes, the statistic average is
equal to the time average. However, for one observation of the correlation function (see
an example in Figure 6.1), typically there is one (or very few) ’peak’ values, and several
out-of-peak values. Thus, the statistical average can be assumed by:


 E(z p ) ≈ z p
N
(6.13)
1

E(z
)
≈
o

N ∑ zo,k
k=1
where N ≥ 1 is the number of out-of-peak values used to estimate the mean. An example
of peak and out-of-peak values based on the non-coherently averaged correlation function
is given in Figure 6.1 with an oversampling factor, NS equals to 4 (for SinBOC(1,1)) or
16 (for BPSK) and modulation order, NB equals to 2 (for SinBOC(1,1)) or 1 (for BPSK).
The modulation order, NB for BOC(n,m) can be given by NB = 2n/m.
6.1.3 PDFs and CDFs
The Probability Distribution Functions (PDFs) of the output of non-coherent integration
in the correct z p and incorrect zo bin hypotheses are given by the non-central and central
χ2 PDFs, respectively [37]:
µ ¶ Nnc −1
fncentr (z p ) =
Nc Nnc
N0
zp
Eeb
2
µ
exp −
(Eeb +z p )Nc Nnc
N0
¶
µ
INnc −1
q
¶
2Nc Nnc z p Eeb
N0
with Iα (x) being an α-order modified Bessel function of first kind, and
µ
¶
zo Nc Nnc
1
N
−1
nc
exp −
zo
fcentr (zo ) = µ
¶Nnc
N0
N0
2Nc Nnc
2Nnc Γ(Nnc )
(6.14)
(6.15)
CNR Estimation
37
Single path static channel
Single path static channel
1
1
Peak values
Out−of−peak values
Peak values
Out−of−peak values
0.9
0.8
0.8
Normalized non−coherent correlation
Normalized non−coherent correlation
0.9
0.7
0.6
0.5
0.4
0.3
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
−4
−3
−2
−1
0
1
Delay[chips]
2
3
0
−4
4
−3
−2
−1
0
Delay[chips]
1
2
3
4
Figure 6.1: Example of peak and out-of-peak values for a non-coherently averaged correlation function in single path static channel. A ’peak’ means a correct bin. SinBOC(1,1)
signal (left) with Nc = 200, Nnc = 1, Ns = 4 and BPSK signal (right) with Nc = 200,
Nnc = 1, Ns = 16, .
with Γ(Nnc ) = (Nnc − 1)! being Gamma (Euler) function.
The Cumulative Distribution Functions (CDFs) for peak and out-of-peak non-coherent
correlation values are [37]:
s
r
µ
¶
2Nc Nnc z p
2Nc Nnc Eeb
Fncentr (z p ) = 1 − QNnc
,
(6.16)
No
No
µ
Fcentr (zo ) = 1 − exp −
zo Nc Nnc
N0
¶ Nnc −1
∑
k=0
µ
¶
1 zo Nc Nnc k
k!
N0
(6.17)
where Qm (a, b) is the generalized Marcum Q function.
6.2
Moment-based CNR estimators
The moment-based CNR estimators are not new in literature. But there has been no
comparison between these moment-based estimators, specially for the GPS/Galileo signal
according to the author’s knowledge. In this thesis, the estimators are derived based
on the autocorrelation function of the incoming signal which is not done before. Some
estimators, namely 1STO, 2NDO-M1, 2NDO-M2, and 4THO-1STO have been the result
of previous, co-authored work in the laboratory [21]. The other estimators are derived
by the author in the similar way and a detailed analysis of all the estimators has been
CNR Estimation
38
performed. An algorithm for choosing the appropriate out-of-peak points is also proposed
in this thesis. The main advantage of the proposed algorithm is that we are able to estimate
the CNR accurately while using lower number of out-of-peak points.
6.2.1
First-order moments (1STO)
From the means given in Equations (6.11) and (6.12), the following two means can be
found:
N0 e
+ Eb
Nc
N0
E(zo ) =
Nc
E(z p ) =
From the second equation, we have, N0 = E(zo )Nc and replacing the value of N0 in the
first equation, we get Eb = E(z p ) − E(zo ). Thus, using the values of N0 and Eb in Equation
(6.1), the following estimate of CNR can be obtained:
d = 10log10 E(z p ) − E(zo ) + 30.
CNR
E(zo )Nc
(6.18)
The mean values, E(z p ), and E(zo ) can be computed by using Equation (6.13).
6.2.2
Second-order moments, method 1 (2NDO-M1)
From the second-order moments given in Equations (6.11) and (6.12), the following two
equations can be found:
µ
¶µ
¶
N0 1
1
2
E(z p ) =
(6.19)
+ 1 2Eeb +
+ Eeb2
Nc Nnc
Nc
E(z2o )
Thus, N02 =
notation x =
µ
¶
N02 1
=
+1
Nc2 Nnc
E(z2p )Nc2
. By replacing this value
1+ N1c
Eeb
N0 , the second-order equation can
(6.20)
in Equation (6.19) above, and using the
be written as:
2 + 1)
E(z2o )(Nnc
E(z2o )Nc2 Nnc 2
x + 2xNc E(z2o ) +
− E(z2p ) = 0.
1 + Nnc
Nnc (1 + Nnc )
(6.21)
CNR Estimation
39
Equation (6.21) has one negative and one positive root. However, SNR should have a
positive value, therefore the positive root is the correct one. Thus, the positive solution of
Equation (6.21) gives the CNR estimate:
sµ
Ã
¶µ
¶!
2)
E(z
p
d = 10log10 − 1+Nnc + 1
(6.22)
CNR
1 + N1nc
+ N1nc
+ 30.
Nc Nnc
Nc
E(z2 )
o
Above, the estimates of the mean squared peak and out-of-peak values can be obtained
via: E(z2p ) ≈ z2p and E(z2o ) ≈
1
N
N
∑ z2o,k .
k=1
6.2.3 Second-order moments, method 2 (2NDO-M2)
Another possibility to estimate CNR is by making the following observation, again based
on Equations (6.11) and (6.12):
√
N0
e N0
E(z p ) − E(zo )
Eeb Nc Nnc
Nc + Eb − Nc
= r
=
(6.23)
std(z2o )
N0
N02
Nnc Nc2
where std(z2o ) =
obtained via:
p
var(z2o ) is the standard deviation. Thus, the CNR estimate can be
p ) − E(zo )
d = 10log10 E(z
√
CNR
+ 30.
Nc Nnc std(zo )
(6.24)
Here, at least two out-of-peak values are needed in order to compute the standard deviation std(z2o ).
6.2.4 Combined second (central) and first (non-central) order moments (2NDO-1STO-M1)
From the second (central) order moments given in Equation (6.11), the following equation
can be found:
µ
¶
N02 1
2
E(zo ) =
+1
(6.25)
Nc2 Nnc
Using Equation (6.25) and the value of E(z p ) from Equation (6.12), it is straightforward to show that an estimate of CNR can be obtained in the form:
µ
¶1
2
2)
E(z
Ã E(z p ) − ¡ o ¢ !
1+ N1nc
d = 10log10
+ 30.
(6.26)
CNR
µ
¶1
2
E(z )
Nc ¡ o1 ¢
1+ Nnc
2
CNR Estimation
40
Again, the approximations, E(z p ) ≈ z p and E(z2o ) ≈
1
N
N
∑ z2o,k can be used.
k=1
6.2.5 Combined second (non-central) and first (central) order moments (2NDO-1STO-M2)
From the second (non-central) order moments given in Equation (6.12), the following
equation is taken:
¶µ
µ
¶
N0 1
1
2
e
+ 1 2Eb +
+ Eeb2
E(z p ) =
Nc Nnc
Nc
(6.27)
Equation (6.27) has one negative and one positive root. However, SNR should have a
positive value, therefore the positive root is the correct one. Thus, the positive solution of
Equation (6.27) gives the Eeb estimate:
sµ
µ
¶
¶2
¶
µ
N02
N0
1
1
1
e
Eb = − 1 + Nnc Nc +
1 + Nnc N 2 − 1 + Nnc NN02 + E(z2p ).
c
c
(6.28)
Again, from the means given in Equation (6.11), the following equation is found:
E(zo ) =
N0
Nc
Therefore, N0 = E(zo )Nc .
The estimated CNR can be obtained in the form:
d = 10log10
CNR
Eeb
+ 30.
Nc E(zo )
(6.29)
where Eeb is given according to the Equation (6.28).
6.2.6
Combined fourth and second order moment (4THO-2NDO)
The fourth order moment of a central ξ2 distribution of variance σ2 and n degrees of
freedom is [1]:
E(z4o ) = nσ8 n(n + 2)(n + 4)(n + 6)
µ
¶µ
¶µ
¶
N04
1
2
3
=
1+
1+
1+
Nc4
Nnc
Nnc
Nnc
(6.30)
CNR Estimation
41
N0 can be written in the following form:
µ
N0 = Nc
E(z4o )
¡
¢¡
¢¡
¢
1 + N1nc 1 + N2nc 1 + N3nc
¶1
4
.
(6.31)
By using Equation 6.31, the estimated CNR can be obtained in the form:
d = 10log10
CNR
Eeb
µ
E(z4o )
1+ N2nc
Nc ¡
¢¡
1+ N1nc
¶ 1 + 30.
¢¡
1+ N3nc
(6.32)
4
¢
where Eeb is again given according to the Equation 6.28.
6.2.7
Combined fourth and first order moments (4THO-1STO)
Using Equation (6.31) and the value of E(z p ) from Equation (6.12), it is straightforward
to show that an estimate of CNR can be obtained in the form:
µ
Ã E(z p )− ¡
d = 10log10
CNR
µ
Nc
¡
1+ N1
nc
1+ N1
nc
E(z4
o)
1+ N2
nc
¢¡
E(z4
o)
1+ N2
nc
¢¡
¢¡
1+ N3
nc
¢¡
1+ N3
nc
¢
¢
¶ 41
!
¶ 14
+30.
Again, the approximations, E(z p ) ≈ z p and E(z4o ) ≈
6.3
(6.33)
1
N
N
∑ z4o,k can be used.
k=1
Simulation results
The common parameters used in the simulation are given in Table 6.1.
Table 6.1: Common parameters for the simulation.
Parameter
Over-sampling rate
Modulation order
Spreading factor
Correlation window
Symbol
Ns
NB
SF
-
Value
4
2
21
7
Unit
Chips
Chips
CNR Estimation
6.3.1
42
Results for single path channel
The results regarding the average errors between the estimated CNR and the true CNR
are shown in Figure 6.2 for single-path static channel, when squared envelope is used as
non-linearity. Small spreading factor (SF = 21) is used in the simulations for the purpose
of increasing the simulation speed. Small differences is expected, if the spreading factor
is increased (e.g., to 1023 chips, as used in GPS/Galileo). In Figure 6.2, the continuous
RMSE
Absolute Mean Error
12
10
9
8
Absolute Mean Error [dB]
7
6
5
2NDO−M2, N=42
2NDO−M1, N=42
1STO, N=42
2NDO−1STO−M2, N=42
4THO−2NDO, N=42
4THO−1STO, N=42
2NDO−M2, N=1
2NDO−M1, N=1
1STO, N=1
4THO−2NDO, N=1
4THO−1STO, N=1
10
8
RMSE [dB]
2NDO−M2, N=42
2NDO−M1, N=42
1STO, N=42
4THO−2NDO, N=42
4THO−1STO, N=42
2NDO−M2, N=1
2NDO−M1, N=1
1STO, N=1
4THO−2NDO, N=1
4THO−1STO, N=1
6
4
4
3
2
2
1
0
15
20
25
30
35
CNR [dBHz]
40
45
50
0
15
20
25
30
35
CNR [dBHz]
40
45
50
Figure 6.2: Examples of CNR estimates for single-path static channel (Absolute Mean
Error and RMSE), Nc = 20 and Nnc = 2.
lines are used to indicate that the CNR computation takes more than one out-of-peak
values (N > 1) in order to estimate E(zo ). All methods use only one value (the global
peak value) to estimate E(z p ). The dashed lines use only 1 out-of-peak value (with the
exception of the method 2NDO-M2 which uses N = 2 out-of-peak samples for estimating
the standard deviation but only 1 point to estimate the statistical averages). Based on
Figure 6.2, all the proposed estimators seem to have similar performance if enough outof-peak values are employed in computing the statistical averages. The performance of all
the methods deteriorates when N decreases, especially at low CNRs (when CNR is high
enough, the number of samples N used in the estimation is not important any more). The
poorer CNR estimates at lower CNR (compared with mid CNRs) can be explained by the
fact that, the lower the CNR is, the less accurate the above formulas are according to [21].
The poorer CNR estimates at high CNR (compared with mid CNRs) can be explained
by some calculus approximations errors that occur in Matlab when input CNR is above a
certain threshold. This can be partly overcome by increasing the number of random points
used in the statistics. In the CNR range of 20 till 45 dB, an RMSE in CNR estimation
CNR Estimation
43
around 2 dB is obtained with all the 7 methods (if N = 42 samples). However with N = 42
samples, the complexity is much higher in comparison with N = 1. So the target is to use
lower number of out-of-peak points and still get low RMSE. From Figure 6.3, it is found
that N ≥ 20 is good to use to get low RMSE.
CNR=40
CNR=25
7
7
2NDO−M2
2NDO−M1
1STO
4THO−1STO
2NDO−1STO−M1
4THO−2NDO
2NDO−1ST−M2
6
6
5
RMSE [dB]
RMSE [dB]
5
4
4
3
3
2
2
1
0
2NDO−M2
2NDO−M1
1STO
4THO−1STO
2NDO−1STO−M1
4THO−2NDO
2NDO−1ST−M2
5
10
15
20
25
30
35
N
40
45
1
0
5
10
15
20
25
30
35
40
45
N
Figure 6.3: RMSE vs. N plots for different CNRs for single-path static channel, Nc = 20
and Nnc = 2.
Choosing the appropriate number of out-of-peak points is an important task as described in the previous paragraph. The following procedure can be used to choose the
appropriate number of out-of-peak points.
• Find the global peak point.
• Find the other peak points which are within NS ∗ NB samples from the main peak in
either direction.
• Mark all other points as out-of-peak points. Suppose NTotal is the total number of
out-of-peak points.
• Compute the mean, MEANPeaks of all the peak points including the global peak.
• Set a threshold, T HRESHOLD = 2.5 ∗ MEANPeaks and compute the number of true
peaks by comparing the the points above the T HRESHOLD. Suppose NPeaks is the
total number of true peaks.
• Compute the appropriate number of out-of-peak points, NAppropriate = NTotal /NPeaks .
CNR Estimation
44
The simulation results using the above procedure for choosing the appropriate out-ofpeak points are shown in Figure 6.7. In this simulation, 20 out-of-peak points are used
on the average. From the results, it is found that 4THO-1STO gives the lowest absolute
mean error and RMSE. Here, Nc = 200 and Nnc = 1 are used, similar to what is used for
processing the measurement data. A plot showing how many out-of-peak points are used
for computing the different CNR levels is given in Figure 6.5. From this figure, it can be
stated that lower number of out-of-peak points are necessary for high CNR estimation in
comparison to the low CNRs.
RMSE
Absolute Mean Error
1.4
2.8
2NDO−M2
2NDO−1STO−M1
1STO
2NDO−1STO−M2
2NDO−M1
4THO−2NDO
4THO−1STO
1.2
2.6
2.4
RMSE [dB]
1
0.8
0.6
2.2
2
0.4
1.8
0.2
1.6
0
20
22
24
26
28
30
32
CNR [dBHz]
34
36
38
2NDO−M2
2NDO−1STO−M1
1STO
2NDO−1STO−M2
2NDO−M1
4THO−2NDO
4THO−1STO
40
1.4
20
22
24
26
28
30
32
CNR [dBHz]
34
36
38
40
Figure 6.4: Examples of CNR estimates for single-path static channel (Absolute Mean
Error and RMSE), Nc = 200 and Nnc = 1.
The estimators were also used to estimate the CNR for single-path fading channel.
Figure 6.6 shows the results for such a channel. The CNRs are estimated in the same way
as the single-path static channel in Figure 6.7. The results for the fading channel is quite
similar with the results for the static channel (i.e., 4THO-1STO shows lowest absolute
mean error and RMSE).
6.3.2 Results for multi-path channel
The CNR estimation for multi-path channels are also studied using the moment-based
estimators. The results are given in Figure 6.7. The same procedure, described in the
previous subsection for choosing the out-of-peak points is used. From this figure, we find
that only two estimators namely, 4THO-1STO and 4THO-2NDO give acceptable RMSE
values around 2.5 − 3 dBHz. These two estimators give around 0.5 − 1 dBHz worse
performance in terms of RMSE for multi-paths compared to single path.
CNR Estimation
45
NAppropriate vs. CNR
20
19.5
19
18
N
Appropriate
18.5
17.5
17
16.5
16
20
22
24
26
28
30
32
CNR [dBHz]
34
36
38
40
Figure 6.5: NAppropriate vs. CNR for single-path static channel, Nc = 200 and Nnc = 1.
RMSE
Absolute Mean Error
1.4
2.8
2NDO−M2
2NDO−1STO−M1
1STO
2NDO−1STO−M2
2NDO−M1
4THO−2NDO
4THO−1STO
1.2
2.6
2.4
RMSE [dB]
1
0.8
0.6
2.2
2
0.4
1.8
0.2
1.6
0
20
22
24
26
28
30
32
CNR [dBHz]
34
36
38
40
2NDO−M2
2NDO−1STO−M1
1STO
2NDO−1STO−M2
2NDO−M1
4THO−2NDO
4THO−1STO
1.4
20
22
24
26
28
30
32
CNR [dBHz]
34
36
38
40
Figure 6.6: Examples of CNR estimates for single-path fading (Nakagami m=0.8) channel
(Absolute Mean Error and RMSE), Nc = 200 and Nnc = 1.
CNR Estimation
46
Absolute Mean Error
RMSE
8
11
2NDO−M2
2NDO−M1
1STO
2NDO−1STO−M2
2NDO−1STO−M1
4THO−2NDO
4THO−1STO
7
9
8
5
RMSE [dB]
6
2NDO−M2
2NDO−M1
1STO
2NDO−1STO−M2
2NDO−1STO−M1
4THO−2NDO
4THO−1STO
10
4
7
6
3
5
2
4
1
3
0
20
22
24
26
28
30
32
CNR [dBHz]
34
36
38
40
2
20
22
24
26
28
30
32
CNR [dBHz]
34
36
38
40
Figure 6.7: Examples of CNR estimates for multi-path (2-4 paths) fading (Nakagami
m=0.8) channel (Absolute Mean Error and RMSE), Nc = 200 and Nnc = 1.
The overall performance of all these estimators can be improved if we are able to
reduce the effects of multi-paths by incorporating multi-path mitigating algorithms. It is
left as future research topic.
6.3.3 Envelope vs. squared envelope as nonlinearity
The moment-based estimators are derived by assuming the squared envelope of the noncoherent blocks. It is also interesting to examine how the estimators perform if the envelope is used instead of the squared envelope as non-linearity. That means the χ2 statistics
are no longer valid. The results are given in Figure 6.8 from where it is easy to find that using of envelope instead of squared envelope gives approximately 5 dB poor performance
in all the methods.
6.4
CNR mappings
An important question is to know how to map the CNR obtained via a certain (Nc , Nnc )
pair to another CNR value, when the integration times are increased. For example, if from
the simulations a good tracking result at CNR equal to CNR1 = 30 dBHz with Nc1 = 10
ms and Nnc1 = 1 block is achieved, the question is: what kind of integration times (Nc2 ,
Nnc2 ) is needed in order to achieve a similar performance at CNR1 = 20 dBHz.
For this, the following equation can be used according to [21]:
CNR Estimation
47
2NDO−M2
2NDO−M1
9
7
Squared Envelope
Envelope
Squared Envelope
Envelope
8
6
7
5
RMSE [dB]
RMSE [dB]
6
5
4
4
3
3
2
2
1
20
22
24
26
28
30
32
CNR [dBHz]
34
36
38
1
20
40
22
24
26
28
30
32
CNR [dBHz]
34
36
38
40
4THO−1STO
1STO
5.5
6
Squared Envelope
Envelope
5.5
Squared Envelope
Envelope
5
5
4.5
4.5
RMSE [dB]
RMSE [dB]
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
20
22
24
26
28
30
32
CNR [dBHz]
34
36
38
1
20
40
22
24
26
28
30
32
CNR [dBHz]
34
36
38
40
2NDO−1STO−M1
4THO−2NDO
6
7
Squared Envelope
Envelope
Squared Envelope
Envelope
5.5
6
5
4.5
RMSE [dB]
RMSE [dB]
5
4
4
3.5
3
3
2.5
2
2
1.5
1
20
22
24
26
28
30
32
CNR [dBHz]
34
36
38
1
20
40
22
24
26
28
30
32
CNR [dBHz]
34
36
38
40
2NDO−1STO−M2
7
Squared Envelope
Envelope
6
RMSE [dB]
5
4
3
2
1
20
22
24
26
28
30
32
CNR [dBHz]
34
36
38
40
Figure 6.8: Examples of CNR estimates for the Envelope Vs. the Squared envelope for
single path fading channel, Nc = 20 and Nnc = 2.
CNR Estimation
CNR2 [dBHz] = CNR1 [dBHz] − 10log10 (Nc1
48
p
Nnc1 ) + 10log10 (Nc2
p
Nnc2 )
(6.34)
For example, if a certain performance at CNR1 = 30 dBHz is achieved with Nc1 = 20
ms and Nnc1 = 1 block, then, in order to achieve the same performance at CNR1 = 10
dBHz, it is needed, for example: Nc2 = 100 ms and Nnc2 = 1 block, or Nc2 = 10 ms and
Nnc2 = 100 blocks, or Nc2 = 5 ms and Nnc2 = 400 blocks, or Nc2 = 4 ms and Nnc2 = 625
blocks. Based on this mapping the authors in [21] have stated that the target should be to
achieve reasonable performance for CNRs as low as 24 dBHz.
As far as the border between indoor/outdoor environments is concerned, there are
several authors in the scientific community who discussed it. Particularly, in [10] the
authors considered this border to be about −155 dBm, or approximately equal to 18 − 19
dBHz. In [12] the indoor environment is characterized by CNR values less than 25 dBHz,
while in [19] it is stated that the indoor acquisition requires successful signal detection at
typically 20 dBHz CNR level.
6.5
CNR estimators results with measurement data
The moment based CNR estimators of Section 6.2 are used to estimate the CNR for the
PL based data as well as for Satellite based data from the normalized CIR amplitudes
computed over Nc = 200. The used PRNs for the satellite data are 3,15,16,18,21 and
26. One snapshot of normalized CIR envelope was given in Figure 5.5 of Chapter 5.
A window of 200 correlators (i.e., about ±6.25 chips around the global peak) was used
(we recall that the oversampling factor was 16.367/1.023 ≈ 16 samples per chip). The
out-of-peak points are the points those were 16 samples outside the main peak in both
directions. A total of 168 out-of-peak points are detected. The same procedure, described
in subsection 6.3.1 for choosing appropriate out-of-peak points is used while estimating
the CNR for measurement data. For the measurement data, 80 out-of-peak points are
used on the average. The CNR estimation standard deviation (std) results are given in the
Table 6.2 (for PLs) and Table 6.3 (for satellites). From the results it is easy to find that
the 4THO-1STO estimator gives the lowest std most of the times. The PL results also
show high CNR std and low mean CNR in MultiplePL in comparison with SinglePL due
to more interference in MultiplePL data as three PLs were used. The results of satellite
data show similar results (i.e. high CNR std in RoomSAT compared to CorridorSAT )
due to less probability of having LOS path in RoomSAT than CorridorSAT . However,
comparing with satellite and PL results, it is possible to find that the PL reference signal
CNR Estimation
remains almost constant (i.e., small std) in comparison with the satellite data over the
captured time as expected. This is due to the fact that the reference receiver was connected
to a PL directly via cable.
6.6
CNR estimation results for 4THO-1STO using different navigation bit estimators
As discussed in the previous section, the 4THO-1STO is the best estimator among the
moment based estimators. Here the different navigation bit estimators of Section 5.3 are
tested in the context of CNR estimation while using the 4THO-1STO method.The navigation bit estimators include different phase thresholds and the signum function approach,
as discussed in Section 5.3. The target is to select the best method based on the lowest
CNR std. From Table 6.4 (for PLs) and Table 6.5 (for satellites), it is easy to find that π is
the best threshold for navigation data bit estimation. So, from the overall results, it can be
stated that 4TH-1STO estimator using π as the threshold to estimate the navigation bits
is the best way to estimate the CNR. The mean CNR using 4THO-1STO estimator for
each data set is given in Table 6.6 (for PLs) and Table 6.7 (for satellites). Here, it should
be mentioned that the estimated CNR varies at most 3 dBHz depending on the different
estimators. The fades in the tables are the differences between the reference signal and
the indoor signal. From the tables, it is found that for PLs the reference signal can have
a mean CNR value approximately equal to 40 − 50 dBHz while indoor signal can have
a mean CNR value of 30 − 40 dBHz. The mean fades in the PLs can be approximately
equal to 5 − 15 dBHz. For satellites, the reference signal’s mean CNR values of 38-42
dBHz are observed while the indoor signal can have a mean CNR value of 19-23 dBHz.
The mean fades in the satellite data can be 18 − 20 dBHz. Estimated CNRs and fades for
a few data sets are shown in Figure 6.9.
6.7 Computational complexity of the estimators
The computational complexity of these estimators have been reduced by using appropriate
out-of-peak points. But still these estimators are complex enough from the computation
point of view. Especially, the estimators with higher order moments (e.g., 4THO-1STO
and 4THO-2NDO) require many exponential operations. The estimator, 4THO-2NDO
shows almost similar results like 4THO-1STO in terms of RMSE for multi-path channels
in the simulation and for measurement based data. The estimators namely, 2NDO-M1,
49
CNR Estimation
50
Table 6.2: CNR standard deviation [dBHz] for different estimators for PL data, Nc = 200.
SET
1
Estimator
1STO
2NDO-M1
2NDO-M2
2NDO-1STO-M1
2NDO-1ST-M2
4THO-1STO
4THO-2NDO
SinglePL
std (Reference, Indoor)
2.59, 3.28
1.20, 1.73
0.92, 1.58
1.15, 1.73
2.65, 3.28
0.40, 1.16
0.42, 1.17
MultiplePL
2.80, 3.18
1.26, 1.75
1.24, 1.73
1.26, 1.75
2.84, 3.18
0.58, 1.48
0.59, 1.49
2
1STO
2NDO-M1
2NDO-M2
2NDO-1STO-M1
2NDO-1STO-M2
4THO-1STO
4THO-2NDO
2.23, 3.93
0.67, 3.05
0.66, 3.02
0.67, 3.04
2.23, 3.94
0.26, 2.86
0.26, 2.87
2.23, 4.26
0.88, 3.50
0.85, 3.59
0.89, 3.17
1.79, 4.27
0.37, 3.18
0.20, 3.18
3
1STO
2NDO-M1
2NDO-M2
2NDO-1STO-M1
2NDO-1STO-M2
4THO-1STO
4THO-2NDO
1.11, 3.93
0.44, 3.48
0.42, 3.42
0.44, 3.47
1.12, 3.94
0.22, 3.43
0.22, 3.44
1.92, 4.26
0.79, 3.83
0.79, 3.76
0.80, 3.82
1.92, 4.34
0.36, 3.54
0.36, 3.55
4
1STO
2NDO-M1
2NDO-M2
2NDO-1STO-M1
2NDO-1STO-M2
4THO-1STO
4THO-2NDO
1.09, 4.25
0.66, 4.09
0.60, 3.94
4.08, 6.66
1.09, 4.30
0.41, 3.84
0.42, 3.85
3.70, 3.62
1.53, 1.82
1.50, 1.77
1.53, 1.82
3.70, 3.61
0.69, 1.36
0.69, 1.36
5
1STO
2NDO-M1
2NDO-M2
2NDO-1STO-M1
2NDO-1STO-M2
4THO-1STO
4THO-2NDO
0.08, 2.05
0.61, 1.50
0.57, 1.35
0.62, 1.44
0.81, 2.11
0.46, 1.28
0.47, 1.33
3.48, 3.57
1.52, 1.97
1.53, 1.77
1.63, 1.87
3.48, 3.51
0.75, 1.28
0.76, 1.30
CNR Estimation
51
Table 6.3: CNR standard deviation [dBHz] for different estimators for satellite data, Nc =
200.
PRN
3
Estimator
1STO
2NDO-M1
2NDO-M2
2NDO-1STO-M1
2NDO-1ST-M2
4THO-1STO
4THO-2NDO
CorridorSAT
1.92, 3.56
1.55, 3.66
1.41, 3.54
1.54, 3.55
1.92, 3.65
1.10, 3.54
1.11, 3.49
RoomSAT
1.81, 2.94
1.79, 2.85
1.43, 2.82
1.72, 2.83
1.94, 2.98
1.14, 2.63
1.19, 2.65
15
1STO
2NDO-M1
2NDO-M2
2NDO-1STO-M1
2NDO-1STO-M2
4THO-1STO
4THO-2NDO
1.64, 2.58
176, 2.68
1.55, 2.68
1.58, 2.67
1.72, 2.59
1.83, 2.30
1.88, 2.35
2.03, 3.02
2.19, 3.04
2.19, 2.99
2.15, 3.01
2.03, 3.03
1.99, 2.47
2.02, 2.55
16
1STO
2NDO-M1
2NDO-M2
2NDO-1STO-M1
2NDO-1STO-M2
4THO-1STO
4THO-2NDO
1.95, 2.29
1.62, 2.26
1.54, 2.15
1.61, 2.18
1.96, 2.36
1.29, 2.15
1.30, 2.25
1.94, 2.37
1.66, 2.25
1.60, 2.21
1.66, 2.22
1.92, 2.42
1.31, 2.19
1.35, 2.26
18
1STO
2NDO-M1
2NDO-M2
2NDO-1STO-M1
2NDO-1STO-M2
4THO-1STO
4THO-2NDO
2.64, 1.51
1.66, 1.85
1.50, 1.65
1.59, 1.85
1.59, 2.64
0.96, 0.97
0.98, 1.02
1.65, 1.86
1.71, 1.85
1.66, 1.72
1.72, 1.92
1.72, 2.48
1.08, 1.28
1.12, 1.22
21
1STO
2NDO-M1
2NDO-M2
2NDO-1STO-M1
2NDO-1STO-M2
4THO-1STO
4THO-2NDO
1.46, 2.09
1.22, 2.14
1.16, 2.02
1.22, 2.04
1.46, 2.18
0.86, 1.98
0.86, 2.06
1.66, 3.11
1.32, 3.15
1.17, 3.03
1.41, 3.16
1.46, 3.21
0.82, 2.94
0.87, 2.95
26
1STO
2NDO-M1
2NDO-M2
2ND-1STO-M1
2NDO-1STO-M2
4THO-1STO
4THO-2NDO
1.00, 1.73
1.07, 1.42
0.93, 1.28
1.00, 1.41
1.17, 1.74
0.78, 1.09
0.83, 1.09
1.73, 1.86
1.63, 1.77
1.33, 1.99
1.73, 1.74
1.72, 1.72
1.37, 1.65
1.40, 1.70
CNR Estimation
52
Table 6.4: 4THO-1STO, CNR standard deviation [dBHz] for PL data, Nc = 200.
SET
1
Method
Threshold
SinglePL
0.97, 1.17
0.40, 1.16
0.63, 1.20
0.73, 1.41
MultiplePL
0.82, 1.63
0.58, 1.48
0.99, 1.60
1.14, 1.90
π/2
π
2π/3
0.45, 3.15
0.26, 2.86
0.97, 3.00
1.35, 3.52
0.79, 3.36
0.37, 3.18
0.87, 3.26
0.98, 4.02
π/2
π
2π/3
0.48, 3.48
0.22, 3.43
0.81, 3.49
1.21, 3.56
0.50, 3.60
0.36, 3.54
0.97, 3.63
1.30, 3.62
π/2
π
2π/3
0.81, 4.01
0.41, 3.84
0.89, 4.00
0.99, 4.25
0.82, 1.63
0.69, 1.36
0.99, 1.60
1.17, 1.90
π/2
π
2π/3
0.81, 1.76
0.46, 1.28
0.99, 1.73
1.41, 2.01
1.05, 1.63
0.75, 1.28
0.97, 1.46
1.17, 1.77
Used thresh.
π/2
π
2π/3
Signum
2
Threshold
Signum
3
Threshold
Signum
4
Threshold
Signum
5
Threshold
Signum
CNR Estimation
53
Table 6.5: 4THO-1STO, CNR standard deviation [dBHz] for satellite data, Nc = 200.
PRN
3
Method
Threshold
CorridorSAT
1.22, 3.62
1.10, 3.54
1.16, 3.59
1.42, 3.76
RoomSAT
1.92, 2.94
1.14, 2.63
1.41, 2.93
1.54, 3.55
π/2
π
2π/3
2.04, 2.66
1.83, 2.30
2.05, 2.70
2.26, 3.17
2.18, 2.67
1.99, 2.47
2.22, 2.86
2.40, 3.19
π/2
π
2π/3
1.47, 2.26
1.29, 2.15
1.48, 2.27
1.97, 2.53
1.47, 2.27
1.31, 2.19
1.41, 2.44
1.56, 2.62
π/2
π
2π/3
1.13, 1.14
0.96, 0.97
1.07, 1.43
1.37, 1.74
1.40, 1.48
1.08, 1.28
1.41, 1.94
1.78, 2.00
π/2
π
2π/3
1.12, 2.07
0.86, 1.98
0.94, 2.16
1.66, 2.40
0.94, 3.18
0.82, 2.94
0.99, 3.09
1.54, 3.55
π/2
π
2π/3
1.09, 1.29
0.78, 1.09
1.00, 1.20
1.63, 1.75
1.45, 1.73
1.37, 1.64
1.41, 1.78
1.54, 2.33
Used thresh.
π/2
π
2π/3
Signum
15
Threshold
Signum
16
Threshold
Signum
18
Threshold
Signum
21
Threshold
Signum
26
Threshold
Signum
Table 6.6: 4THO-1STO, mean CNR [dBHz] for PL data, Nc = 200.
SET
1
2
3
4
5
SinglePL
Reference Indoor
46.32
41.66
47.99
41.05
45.57
39.20
49.76
40.29
49.12
40.52
Fade
4.66
6.94
6.37
11.47
10.60
MultiplePL
Reference Indoor
43.59
31.35
46.00
34.21
45.96
34.19
44.34
30.72
42.34
31.22
Fade
12.24
11.79
11.77
13.62
11.12
CNR Estimation
54
Table 6.7: 4THO-1STO, mean CNR [dBHz] for satellite data, Nc = 200.
PRN
3
15
21
18
21
26
CorridorSAT
Reference Indoor
41.68
22.96
39.40
20.50
42.31
23.10
38.94
20.18
39.85
21.74
42.04
23.71
SET−2, SinglePL, Mean CNR (Ref, Indoor) =[47.9875
Fade
18.20
18.90
19.21
18.76
18.11
18.33
Fade
19.37
20.67
19.70
20.70
20.73
18.64
SET−2, MultiplePL, Mean CNR (Ref, Indoor) =[46.0074
41.0507] dB
50
34.2083] dB
50
Ref
Indoor
Fades
45
Ref
Indoor
Fades
45
40
40
35
CNR estimates (dB−Hz)
RoomSAT
Reference Indoor
39.94
20.57
40.29
19.62
40.46
20.76
40.15
19.45
39.88
19.15
39.80
21.16
30
25
20
15
35
30
25
20
15
10
10
5
0
0
20
40
60
80
100
5
0
120
20
40
Time (sec)
60
80
100
Time (sec)
PRN 15, CorridorSAT, Mean CNR (Ref, Indoor) =[39.3967
PRN 15, RoomSAT,
20.5097] dB
45
Mean CNR (Ref, Indoor) =[40.2955
19.6254] dB
50
Ref
Indoor
Fades
40
Ref
Indoor
Fades
45
40
35
30
25
35
30
25
20
20
15
15
10
0
10
10
20
30
Time (sec)
40
50
60
5
0
10
20
30
40
50
60
Time (sec)
Figure 6.9: Examples of CNR estimates and the fades for PLs (upper) and satellite (lower)
data, Nc = 200 ms.
CNR Estimation
2NDO-1STO-M1 and 2NDO-1STO-M2 have lower computational complexity compared
to the estimators using higher order moments. The simulation shows that 2NDO-1STOM1 gives the best results among these three estimators. From the measurement based
data same conclusion has been obtained. The most computational efficient estimators are
1STO and 2NDO-M2. But these estimators are not best in terms of RMSE. However, for
single-path case, 1STO shows quite acceptable results in terms of RMSE. If RMSE (for
simulation), std (for measurement data) and computational complexity are considered,
then 2NDO-1STO-M1 will be the best estimator. However, in this thesis, the author has
chosen the best estimator in terms of RMSE (for simulation) and std(for measurement
data).
55
Chapter 7
Channel models based on measurement
data
This chapter discusses channel modeling based on the measurement data obtained in the
measurement campaigns described in Chapter 4. The fading distribution of the indoor
signal is compared with the Rayleigh [48], Rice [24, 48], Nakagami-m [27, 49], Log
Normal [48], and Loo [31] discussed in Chapter 2 and combined distributions discussed
in the first section of this chapter. This chapter first presents the proposed fading channel
model. Then, it discusses the results of fading distribution matching. Following that, it
discusses the number of paths and time dispersion parameters of the indoor signals. This
type of results were previously presented in [20] and [26] for a total of 4 sets of PL data
and 2 sets of satellite data, respectively. More detailed analysis has been performed by
using more PL data as well as satellite data in this thesis.
7.1
Proposed combined fading channel models
According to [20], no fading distribution matches fully with the PL data. So it is interesting to check the combined distributions also for better match. The following models are
developed consisting of weighted distribution of Nakagami, Log-normal, Rayleigh and
Loo distributions.


 fCombined−M1 (w) = C fNaka (w) + D fLogn (w) + E fRayl (w)
fCombined−M2 (w) = E fNaka (w) + D fLogn (w) +C fRayl (w)
p
p


fCombined−M3 (w) = (D2 − E 2 ) fLoo (w) + (C2 + D2 ) fLogn (w)
(7.1)
where C, D and E are the weighting factors representing the probability of having
56
57
single path, 2-3 paths and more than 3 paths, respectively, and fRayl , fLogn , fNaka , fLoo
are the Rayleigh, Log-normal, Nakagami-m and Loo PDFs, respectively. The weighting
factors are estimated from the measurement data by computing the true channel peaks of
CIR amplitudes. All the peaks above a threshold of 2.5 ∗ mean(CIR) are considered as
the true channel peaks. Example Plots of combined distributions are shown in Figure 7.1.
Combined PDF
0.7
Combined−M1
Combined−M2
Combined−M3
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
Figure 7.1: Different combined distributions
7.2
Fading distribution matching
7.2.1
Pseudolite results
The best fit with theoretical distributions (of the instantaneous and averaged amplitude
distributions) is searched by minimizing the Mean Square Error (MSE) between measurement data CDF and theoretical distribution CDF. The probability that the envelope of
the received signal does not exceed a specified value, W is given by corresponding CDF
and can be defined in terms of PDF in the following way [39]:
F(W ) =
Z W
0
f (w)dw
(7.2)
The CDF is computed for each theoretical distribution as well as for the measurement
data distribution. The best theoretical distribution is then chosen based the minimum
MSE between the theoretical CDF and the measurement data CDF. The LOS delay is
computed in two ways: either as the delay corresponding to the peak of the reference
signal bτLOS,re f (using the fact that the two receivers are synchronized), or as the delay of
58
the global peak in the wireless signal bτ peak,air . According to the minimization criterion,
the results are shown in Table 7.1 and Table 7.2 for different PL data. The notations, M1,
M2 and M3 represent the three combined PDFs discussed in the previous section. The
tables show both best amplitude distribution fits at bτLOS,re f and bτ peak,air . A p sign is added
in the tables to present best amplitude distribution at bτ peak,air . For example, (Rayl, M1 p )
means that Rayleigh is the best fit at bτLOS,re f and M1 is the best fit at bτ peak,air . The first
row of these tables shows the best fit for instantaneous amplitude distribution (Nc = 1),
and the last two rows show the best fit for averaged amplitude distribution, for Nc = 50
and Nc = 200 ms coherent integration.
Table 7.1: Best amplitude distribution fit at bτLOS,re f and bτ peak,air , SinglePL
Nc
1 ms
50 ms
200 ms
SET 1
Rayl, Rayl p
M1, M1 p
M1, M1 p
SET 2
Rayl, Rice p
M1, M1 p
M1, M1 p
SET 3
M1, M1 p
M1, M1 p
M1, M1 p
SET 4
M1, M1 p
M1, M1 p
M1, M1 p
SET 5
M1, M1 p
M1, M1 p
M1, M1 p
Table 7.2: Best amplitude distribution fit at bτLOS,re f and bτ peak,air , MultiplePL
Nc
1 ms
50 ms
200 ms
SET 1
Rayl, Logn p
M2, M2 p
M2, M2 p
SET 2
Rayl, M2 p
M2, M2 p
M2, M2 p
SET 3
Rayl, Rice p
M2, M2 p
M2, M2 p
SET 4
M2, Logn p
M2, M2 p
M2, M2 p
SET 5
M2, Rayl p
M2, M3 p
Logn, M2 p
It is observed that for SinglePL, the best distribution for the instantaneous amplitudes is surely Combined-M1 distribution with a few exceptions. For averaged amplitudes
(Nc = 50 and Nc = 200) of SinglePL, still Combined-M1 is the best fit. For MultiplePL,
Combined-M2 distribution is the best match with some exceptions. The MSE between
theoretical and measured distributions are shown for SET-1, SinglePL in Table 7.3 and
Table 7.4 for the distribution matching at bτLOS,re f and at bτ peak,air , respectively. The corresponding few PDF matching plots are shown in Figure 7.2. The upper and lower plots are
for the distribution matching at bτLOS,re f and bτ peak,air respectively.
7.2.2
Satellite results
The best fit with theoretical distributions (of the instantaneous and averaged amplitude
distributions) of the satellite data for PRN 3, 15, 16, 18, 21 and 26 are searched in similar
59
Table 7.3: MSE between the theoretical and measured distributions at bτLOS,re f , SET-1,
SinglePL
Nc
1 ms
50 ms
200 ms
Rayl
0.0056
0.0316
0.0328
Rician
0.0060
0.0354
0.0390
Naka-m
0.0107
0.0099
0.0142
Loo
0.0057
0.0354
0.0390
Logn
0.0347
0.0346
0.0258
M1
0.0089
0.0075
0.0120
M2
0.0062
0.0271
0.0275
M3
0.0340
0.0341
0.0254
Table 7.4: MSE between the theoretical and measured distributions at bτ peak,air , SET-1,
SinglePL
Nc
1 ms
50 ms
200 ms
Rayl
0.0101
0.0305
0.0322
Rician
0.0077
0.0344
0.0381
Naka-m
0.0145
0.0108
0.0159
Loo
0.0080
0.0343
0.0381
Logn
0.0248
0.0299
0.0217
Amplitude distribution, indoor channel, Nc=1 ms
0.08
0.07
0.06
0.8
0.7
0.6
0.05
0.5
0.04
0.4
0.03
0.3
0.02
0.2
0.01
0.1
0.5
1
1.5
2
0
0
2.5
Peak amplitude
Rayleigh
Rice.
Nakagami−m.
Loo
Log−normal
Combined−M1
Combined−M2
Combined−M3
Measurements
0.9
CDF
Normalized PDF
M3
0.0245
0.0295
0.0213
1
Rayleigh
Rice.
Nakagami−m.
Loo
Log−normal
Combined−M1
Combined−M2
Combined−M3
Measurements
0.09
0.5
1
1.5
2
2.5
Peak amplitude
4
x 10
4
x 10
Amplitude CDF, indoor channel, Nc=1 ms
0.12
1
Rayleigh
Rice.
Nakagami−m.
Loo
Log−normal
Combined−M1
Combined−M2
Combined−M3
Measurements
0.1
Rayleigh
Rice.
Nakagami−m.
Loo
Log−normal
Combined−M1
Combined−M2
Combined−M3
Measurements
0.9
0.8
0.7
0.6
CDF
0.08
Normalized PDF
M2
0.0087
0.0265
0.0277
0.1
0
0
M1
0.0130
0.0083
0.0133
0.06
0.5
0.4
0.04
0.3
0.2
0.02
0.1
0
0
0.5
1
1.5
Peak amplitude
2
2.5
3
4
x 10
0
0
0.5
1
1.5
Peak amplitude
2
2.5
3
4
x 10
Figure 7.2: PDF and CDF matching based on wireless signal instantaneous amplitude at
bτLOS,re f (upper) and bτ peak,air (lower), SET-1, SinglePL, Nc = 1 ms.
60
way as for PLs and the results are shown in Tables 7.5 and 7.6. The same type of table
representations are used as for PLs (i.e., the meaning of p is same as for PLs).
Table 7.5: Best amplitude distribution fit at bτLOS,re f and bτ peak,air , CorridorSAT
Nc
1 ms
50 ms
200 ms
PRN 3
M2, M2 p
M1, Logn p
M3, Logn p
PRN 15
M2, M3 p
M2, M3 p
M2, M3 p
PRN 16
Rayl, M3 p
M3, M3 p
M3, M3 p
PRN 18
M2, M3 p
M2, M3 p
M2, M3 p
PRN 21
M2, M3 p
M2, M3 p
Rayl, M3 p
PRN 26
M2, M3 p
M2, M3 p
M2, Naka p
Table 7.6: Best amplitude distribution fit at bτLOS,re f and bτ peak,air , RoomSAT
Nc
1 ms
50 ms
200 ms
PRN 3
M2, M3 p
M3, M3 p
M3, M3 p
PRN 15
M2, M3 p
M3, M3 p
Naka, M3 p
PRN 16
M2, M3 p
M3, M3 p
M3, M3 p
PRN 18
M2, M3 p
M3, M3 p
M1, M3 p
PRN 21
M2, M3 p
M3, M3 p
M1, M3 p
PRN 26
M2, M3 p
M3, M3 p
M3, M3 p
For the CorridorSAT , the best distribution fit corresponding to bτLOS,re f is CombinedM2 distribution and the best distribution fit corresponding to bτ peak,air is Combined-M3
distribution. For RoomSAT , it is seen that the best fading distribution is Combined-M3
for both bτLOS,re f and bτ peak,air . An example of PDF matching with theoretical distribution
is shown in Figure 7.3 for PRN 21, CorridorSAT . The upper and lower plots are for the
distribution matching at bτLOS,re f and bτ peak,air , respectively.
The combined distributions do not seem good enough for satellites, in the sense that
they depend on the environment. That is why, the ’next best fit’, meaning the best matching among all the other distributions, except the combined ones are presented in Table
7.7 and Table 7.8 for CorridorSAT and RoomSAT , respectively. From these tables, it is
seen that no distribution matches perfectly with the measured distribution. So, it can be
stated that combined distributions offer some degree of good match, but no distribution
fits perfectly with the satellite data.
Table 7.7: The ’next best fit’ amplitude distribution fit at bτLOS,re f and bτ peak,air ,
CorridorSAT
Nc
1 ms
50 ms
200 ms
PRN 3
Rayl, Logn p
Logn, Logn p
Logn, Logn p
PRN 15
Rayl, Logn p
Rayl, Logn p
Naka, Naka p
PRN 16
Rayl, Rayl p
Logn, Logn p
Logn, Logn p
PRN 18
Rayl, Logn p
Rayl, Logn p
Naka, Naka p
PRN 21
Rayl, Logn p
Rayl, Logn p
Rayl, Logn p
PRN 26
Rayl, Naka p
Rayl, Naka p
Naka, Naka p
61
Amplitude distribution, indoor channel, N =1 ms
Amplitude CDF, indoor channel, N =1 ms
c
c
0.2
1
Rayleigh
Rice.
Nakagami−m.
Loo
Log−normal
Combined−M1
Combined−M2
Combined−M3
Measurements
0.18
0.16
0.12
0.8
0.7
0.6
CDF
Normalized PDF
0.14
0.1
0.5
0.08
0.4
0.06
0.3
0.04
0.2
0.02
0.1
0
0
500
1000
1500
Rayleigh
Rice.
Nakagami−m.
Loo
Log−normal
Combined−M1
Combined−M2
Combined−M3
Measurements
0.9
0
0
2000
500
Peak amplitude
1500
2000
0.25
1
Rayleigh
Rice.
Nakagami−m.
Loo
Log−normal
Combined−M1
Combined−M2
Combined−M3
Measurements
0.2
0.15
0.1
Rayleigh
Rice.
Nakagami−m.
Loo
Log−normal
Combined−M1
Combined−M2
Combined−M3
Measurements
0.9
0.8
0.7
0.6
CDF
Normalized PDF
1000
Peak amplitude
0.5
0.4
0.3
0.05
0.2
0.1
0
0
500
1000
1500
2000
2500
0
0
500
Peak amplitude
1000
1500
2000
2500
Peak amplitude
Figure 7.3: PDF and CDF matching based on wireless signal instantaneous amplitude at
bτLOS,re f (upper) and bτ peak,air (lower), PRN 21, CorridorSAT , Nc = 1 ms.
Table 7.8: The ’next best fit’ amplitude distribution fit at bτLOS,re f and bτ peak,air , RoomSAT
Nc
1 ms
50 ms
200 ms
PRN 3
Rayl, Logn p
Logn, Logn p
Logn, Logn p
PRN 15
Rayl, Logn p
Rayl, Logn p
Naka, Logn p
PRN 16
Rayl, Rayl p
Logn, Logn p
Logn, Logn p
PRN 18
Rayl, Logn p
Rayl, Logn p
Naka, Logn p
PRN 21
Rayl, Logn p
Rayl, Logn p
Naka, Logn p
PRN 26
Rayl, Naka p
Rayl, Naka p
Naka, Naka p
7.3
Average path number and time dispersion parameters
The number of paths is searched from the indoor CIR estimates over 200 ms. The local
peaks of CIR amplitudes are detected and all the local peaks higher than a certain threshold are considered as true channel peaks. The chosen threshold is 2.5 ∗ mean(CIR). The
scaling factor in the threshold is chosen according to the trial and error basis with the
target of estimating correct number of paths in the indoors. Also, in order to compare
different multi-path channels, time dispersion parameters such as the mean excess delay
and RMS delay spread discussed in Chapter 2 are computed for each set of measurement
data.
7.3.1 Pseudolite results
For the PLs, one example CIR snapshot with the specified threshold is shown in Figure
7.4. The number of peaks is computed from this type of CIRs and its mean, maximum and
standard deviation are shown in Table 7.9. The table shows that there can be 1-2 paths
for PL signals. The MultiplePL contains more paths than SinglePL due to the use of
multiple PLs. It is observed from CNR estimation results of Chapter 6 that MultiplePL
signal is noisier than SinglePL and can be verified by checking higher mean, standard
deviation and maximum values of the paths from the table. It is also observed from the
simulation results that in the PL data, only one path is detected most of the times with
a few exceptions of detecting two or more paths. The average, maximum, and standard
deviation of the successive path spacing, when more than 1 path is present, are given in
Table 7.10. Clearly, if more that 1 path is present, the successive paths are spaced at very
short distance from the first one. Also, the RMS delay spread and the mean excess delay
are given in Table 7.11. Here, it is needed to mention that one chip delay means 977.5 ns
and corresponds to 293.25 m error distance. RMS delay spreads of few meters are noticed
which are as expected in such indoor scenarios.
7.3.2 Satellite results
For the satellites, one example CIR snapshot with the specified threshold is shown in
Figure 7.5. The number of peaks is computed from this type of CIRs and its mean,
maximum and standard deviation are shown in Table 7.12. The table shows that there
can be 1-2 paths for satellite signals also. However, the RoomSAT contains more paths
than CorridorSAT due to more multi-paths in rooms. It is also observed from the results
62
63
CIR envelope
1
Ref
Indoor
threshold
0.5
0
−8
−6
−4
−2
0
2
4
Delay error [chips]
6
8
CIR envelope
1
Ref
Indoor
threshold
0.5
0
−8
−6
−4
−2
0
2
Delay error [chips]
4
6
8
Figure 7.4: Example snapshots of CIR envelope and threshold, SET-5, SinglePL.
Table 7.9: Estimated number of channel paths (indoor channel) for PL
SinglePL
MultiplePL
SET mean std max mean std max
1
1.17 0.43
5
1.16 0.40
4
2
1.14 0.41
4
1.12 0.60
6
3
1.07 0.43
6
1.18 0.68
5
4
1.04 0.19
2
1.20 0.56
5
5
1.07 0.29
3
1.18 0.62
4
Table 7.10: Successive path spacing (indoor channel) for PL in [chips].
SinglePL
MultiplePL
SET mean std max mean std max
1
0.10 0.30 1.43 0.03 0.14 1.37
2
0.12 0.36 1.16 0.07 0.08 1.31
3
0.03 0.19 1.19 0.04 0.10 1.85
4
0.03 0.17 1.38 0.06 0.15 1.91
5
0.05 0.22 1.37 0.07 0.21 1.75
64
Table 7.11: Average RMS delay spread and average excess delays for PL.
SinglePL
MultiplePL
SET RMS [ns] Excess delay [ns] RMS [ns] Excess delay [ns]
1
0.65
2.19
14.84
34.01
2
0.51
2.07
34.30
81.97
3
15.35
43.98
36.29
92.22
4
10.33
27.08
20.93
70.12
5
7.30
20.60
34.22
79.12
of the analysis that the probability of detecting single path in CorridorSAT data set is
slightly higher than that of RoomSAT . Sometimes up-to 6 paths are noticed in both of the
data sets. The average, maximum, and standard deviation of the successive path spacing,
when more than 1 path is present, are given in Table 7.13. This table shows that the paths
are very closely spaced for both satellite data sets, if more than 1 path is present. Again,
RMS delay spread and the mean excess delay are given in Table 7.14. The computed
RMS and mean excess delay are much higher than the PL results. However, the values
are still within one chip and gives few tens of meters of error distance.
CIR envelope
1
Reference
Received
threshold
0.5
0
−8
−6
−4
−2
0
2
4
Delay error [chips]
Snapshot of CIR envelope, N =200ms
6
8
c
CIR envelope
1
Reference
Received
threshold
0.5
0
−8
−6
−4
−2
0
2
Delay error [chips]
4
6
8
Figure 7.5: Example snapshots of CIR envelope and threshold, PRN 3, CorridorSAT .
Table 7.12: Estimated number of channel paths (indoor channel) for satellites
CorridorSAT
RoomSAT
PRN mean std max mean std max
3
1.28 0.53
3
1.35 0.64
5
15
1.22 0.48
4
1.20 0.47
3
16
1.26 0.58
4
1.34 0.66
5
18
1.39 0.76
5
1.38 0.79
6
21
1.19 0.46
3
1.32 0.62
4
26
1.2
0.5
4
1.31 0.71
5
Table 7.13: Successive path spacing (indoor channel) for satellites in [chips].
CorridorSAT
RoomSAT
PRN mean std max mean std max
3
0.19 0.66 6.43 0.05 0.29 3.81
15
0.24 1.3 7.16 0.20 0.91 6.12
16
0.13 0.7 6.19 0.09 0.44 4.93
18
0.22 1.13 7.93 0.15 0.70 5.60
21
0.17 0.95 9.9 0.19 0.85 6.81
26
0.10 0.68 8.5 0.10 0.48 5.91
Table 7.14: Average RMS delay spread and average excess delays for satellites.
CorridorSAT
RoomSAT
PRN RMS [ns] Excess delay [ns] RMS [ns] Excess delay [ns]
3
78.89
189.30
26.24
62.92
15
74.49
239.80
70.16
218.00
16
62.68
162.11
40.54
109.97
18
88.82
219.00
74.21
172.60
21
70.03
167.91
82.64
215.98
26
47.39
133.50
77.5
172.22
65
7.4
Comparison between pseudolites and satellite results
From the results of the PLs and satellites, it was observed in Chapter 6 that the indoor
CNR of PLs was higher than that of satellites. In the current chapter, it is also observed
that higher number of paths are available in the satellite signals in comparison to the PLs.
However, more closely spaced paths are observed in the PLs. But in both PLs and satellite
data, the path spacing is less than one chip. The RMS delay and mean excess delay for
the satellite data is higher than those of the PLs, as expected. But still these values are
within one chip.
66
Chapter 8
The conclusions of the work, some discussions, and possible future research topics are
presented in this chapter. Even though it is impossible to make one final conclusion due
to the wide topic area of the thesis, all the main results are presented in this chapter in
brief manner.
8.1
Conclusions
Based on the simulation based BOC modulated signal and measurement based BPSK
signal from the PLs and satellites, an analysis of the CNR estimators has been performed
for the purpose of identifying the most robust CNR estimator. It was noticed that the
proposed 4THO-1STO estimation method showed the lowest RMSE to estimate the CNR
for the simulation based SinBOC(1,1) modulated signal. It was also noticed that 4THO1STO was the most robust in the GPS based PLs and satellites signals.
Based on measurement data of PLs and using the GPS C/A signal, an analysis of
the indoor fading channel model was also performed for the purpose of deriving a good
channel model for GPS signals in the typical indoors. It was noticed that the amplitude
variations can be best modeled by the proposed Combined-M1 (combination of Nakagami, Log-normal and Rayleigh) distribution for single PL measurements. For multiple
PLs measurements, the proposed Combined-M2 (combination of Nakagami, Log-normal
and Rayleigh with different weighting factors than Combined-M1) was the best fit. It was
also noticed that typical indoor channels have few paths spaced at short distances (i.e., less
than 1 chips). The RMS delay spreads were also within one chip, or few hundred meters.
For the satellite measurements, combined distributions offer some degree of good match,
but no distribution fits perfectly with the satellite data. In the corridors, it was noticed that
the indoor signal’s amplitude variations corresponding to the peak of the reference signal
67
can be best modeled by Combined-M2 distribution. However, for the satellite measurements in the office rooms, it was noticed that the amplitude variations can be best modeled
by Combined-M3 (combination of Loo and Log-normal) distribution.
8.2
Future Works
In this thesis, the focus was on SinBOC(1,1) as the candidate BOC modulation. However,
this work can be extended as analyzing the performance of these estimators for other
representative BOC modulation such as CosBOC(15,2.5), CosBOC(10,5) and especially
MBOC that has been chosen as the modulation technique for the future Galileo OS signal
at L1 frequency, and also by GPS for its modernized L1C (i.e., L1 Civil) signal in the
recent studies.
The future research also focuses on the update the moment-based estimators by updating the assumptions of the statistical averages of the peak and out-of-peak points to
achieve better CNR estimates.
Finally, although the CNR estimators are tested for GPS based measurement data, it
would be beneficial to test the estimators in the measured Galileo data and also derive
the channel model for measurement based Galileo signals, provided that Galileo signals
become available from the sky.
68
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73
Appendix: Phase variation and Delay
error estimation
Phase variation
The estimation of the phase between the I and Q components of the correlator output
is carried out for all the measurement data and few snapshots for few data sets are shown
in Figure A.1. The measured phase of the received signal is analyzed over the course of
the data sets to detect any instances of severe phase change, which may indicate changing
multi-path conditions. The figure shows the phase estimates only for few seconds for better understanding. Theoretically, the phase shape should remain relatively constant over a
period multiple of 20 ms in the presence of navigation data. In the absence of navigation
data, the mean of the phase should be around 0. The phase estimates shown in the figure
are in the absence of the navigation data. But still, some phase changes are found. Any
change in the shape can be explained by the presence of the multi-path propagation.
The carrier phase measurements are useful for LOS estimation if there is a clear single
path. But, the multi-paths are mostly available in the indoors and it can be considered as
a major source of errors. Several techniques can be found from the literature for dealing
with the problem of carrier phase multi-paths. An well-known technique for mitigating
the carrier phase multi-paths is the Multi-path Estimating Delay Lock Loop (MEDLL)
which detects and removes the multi-paths. But, its implementation is very complex.
Another technique is found in the literature based on the electromagnetic modeling that
can model and remove errors in the carrier phase observation.
74
Appendix A: Phase variation and Delay error estimation
SET−1, SinglePL,
75
SET−1, MultiplePL,
Estimated phase at global peak
4
1.5
1
3
0.5
2
0
Phase estimation
Phase estimation
1
0
−1
−0.5
−1
−1.5
−2
−2
−3
−2.5
−4
0
50
100
150
−3
0
200
50
Time (msec)
PRN 16, CorridorSAT,
100
150
200
Time (msec)
PRN 16, RoomSAT,
2
1
0.5
1
−0.5
Phase estimation
Phase estimation
0
0
−1
−1
−1.5
−2
−2
−3
−2.5
−4
0
200
400
600
Time (msec)
800
1000
−3
0
200
400
600
800
1000
Time (msec)
Figure A.1: Phase estimates of the indoor signals for PLs (upper plots) and satellites
(lower plots), Nc = 1 ms.
Appendix A: Phase variation Delay error estimation
76
Delay error estimation
The delay of the global peak based on the indoor signal is compared with the delay of
the global peak from the reference signal, in order to find out more about the LOS error
distribution. The histograms of the peak delay differences are illustrated in Figure A.2.
The upper plots of the figure are for PLs while the lower two plots are for satellites. Only
four example plots are shown for four data sets. From the PL plots, it is possible to find
that most of the times either there is 0 LOS delay error or very short delay error within
one chip. But for the satellites data, although less than one chip delay error is found most
of the times, still there is some possibility of getting longer delay error. Similar types of
results were obtained for other PL sets and satellite PRNs.
SET−1, SinglePL,
SET−1, MultiplePL, Histogram of delay error
Histogram of delay error
80
45
70
40
35
60
30
Histogram (%)
Histogram (%)
50
40
30
25
20
15
20
10
10
0
−4
5
−3
−2
−1
0
1
2
3
0
−4
4
Delay difference (indoor peak−reference peak) [chips]
−2
−1
0
1
2
3
4
3
4
Delay difference (indoor peak−reference peak) [chips]
PRN 16, RoomSAT, Histogram of delay error
50
50
45
45
40
40
35
35
30
30
25
20
25
20
15
15
10
10
5
0
−4
−3
Histogram of delay error
Histogram (%)
Histogram (%)
PRN 16, CorridorSAT,
5
−3
−2
−1
0
1
Delay difference (Indoor peak−Reference peak) [chips]
2
3
4
0
−4
−3
−2
−1
0
1
2
Delay difference (Indoor peak−Reference peak) [chips]
Figure A.2: LOS delay error based on the comparison between reference and air signals
for PLs (upper plots) and satellites (lower plots), Nc = 200 ms.

Master of Science Thesis

Transcription

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