\addchap{Dictionary of acronyms}
%\chapter*{Dictionary of acronyms}
\begin {table}[ht]
%\caption{Example uncertainties (latitude and longitude) for various integer values of $K$}
\label{tab:dctionary}\centering 
\fontfamily{iwona}\selectfont
  \begin{tabular}{ll}
%\toprule 
%$D$&&$P_u$&$\sigma_N$\\
\textbf{Acronym} & \textbf{Description}\\\toprule
AGCH&Access Grant Channel\\\midrule
AGPS&Assisted GPS\\\midrule
AOA&Angle of Arrival\\\midrule
ASN.1&Abstract Syntax Notation One\\\midrule
AUC&Authentication Center\\\midrule
ARFCN&Absolute Radio Frequency Channel Number\\
\\
BCCH&Broadcast Common Control Channel\\\midrule
BPSK&Binary Phase Shift Keying\\\midrule
BSC&Base Station Controller\\\midrule
BSS&Base Station Subsystem\\\midrule
BTS&Base Transceiver Station\\
\\
C/A&Code/Acquisition\\\midrule
CBCH&Cell Broadcast Channel\\\midrule
CCH&Controlling/Signalling Channels\\\midrule 
CDMA&Code Division Multiple Access\\\midrule
CPU&Central Processing Unit\\
\\
DGPS&Differential GPS corrections\\
\\
E-OTD&Enhanced Observed Time Difference\\\midrule
EIR&Equipment Identity Register\\\midrule
ETSI&European Telecommunications Standards Institute\\

\end {tabular}
\end {table}

\begin {table}[ht]
%\caption{Example uncertainties (latitude and longitude) for various integer values of $K$}
\label{tab:dctionary}\centering 
\fontfamily{iwona}\selectfont
  \begin{tabular}{ll}
%\toprule 
%$D$&&$P_u$&$\sigma_N$\\
\textbf{Acronym} & \textbf{Description}\\\toprule
FACCH&Fast Associated Control Channel\\\midrule
FCC&US Federal Communication Commission\\\midrule
FCCH&Frequency Correction Channel\\\midrule
FDMA&Frequency Division Multiple Access\\
\\
GMSC&Gateway Mobile Switching Center\\\midrule
GNSS&Global Navigation Satellite System\\\midrule
GPS&Global Positioning System\\\midrule
GSM&Global System for Mobile Communications\\
\\
HLR&Home Location Register\\\midrule
HOW&Handover word\\
\\
IE&Information Element\\\midrule
IMSI&International Mobile Subscriber Identity\\
\\
LBS&Location-Based Service\\\midrule
LFSR&Linear Feedback Shift Registers \\\midrule
LMU&Location Measurement Units\\\midrule
LTE&Long Term Evolution\\
\\
MAC&Media Access Control Address\\\midrule
MS&Mobile Station\\\midrule
MSC&Mobile Switching Center\\\midrule
MSISDN&Mobile Subscriber Integrated Services Digital Network-Number\\
\\
NSS&Network Switching Subsystem\\\midrule
NVCS&Navigation Center of the US Coast Guard\\
\\
PCH&Paging Channel\\\midrule
PDU&Protocol Data Unit\\\midrule
PER&Packed Encoding Rules\\\midrule
PLL&Phase Locked Loop\\\midrule
PRN&Pseudo Random Noise
\end {tabular}
\end {table}

\newpage
\begin {table}[ht!]
%\caption{Example uncertainties (latitude and longitude) for various integer values of $K$}
\label{tab:dctionary}\centering 
\fontfamily{iwona}\selectfont
  \begin{tabular}{ll}
%\toprule 
%$D$&&$P_u$&$\sigma_N$\\
\textbf{Acronym} & \textbf{Description}\\\toprule
RACH&Random Access Channel\\\midrule
RF&Radio Frequency\\\midrule
RINEX&Receiver Independent Exchange Format\\\midrule
RRLP&Radio Resource Location Protocol\\\midrule
RSS&Received Signal Strength\\
\\
SACCH&Slow Associated Control Channel\\\midrule
SCH&Synchronization Channel\\\midrule
SDCCH&Stand-alone Dedicated Control Channel\\\midrule
SDR&Software Defined Radio\\\midrule
SIM&Subscriber Identification Module\\\midrule
SMLC&Serving Mobile Location Center\\\midrule
SMS&Short Message Services\\
\\
TCH&Traffic Channels\\\midrule
TDMA&Time Division Multiple Access\\\midrule
TLM&Telemetry\\\midrule
TRAU&Transcoding Rate, Adaptation Unit\\\midrule 
TS&Telecommunication Standard\\
\\
UL-TDOA&Up-Link Time Difference of Arrival\\\midrule
UMTS&Universal Mobile Telecommunications System\\\midrule
USRP&Universal Software Radio Peripheral\\\midrule
UTC&Coordinated Universal Time\\
\\
VLR&Visitor Location Register\\\midrule
VTY&Virtual Teletype
\\\bottomrule
\end {tabular}
\end {table}

\addchap{Appendix}
\numberwithin{equation}{section}
\numberwithin{table}{section}
%\captionsetup[figure]{list=no}
%\captionsetup[table]{list=no}
\numberwithin{figure}{section}

\section{Installation and configuration guide}
\label{sec:instConf}
In order to evaluate the localization system, it is required to install OpenBSC
and to modify the proper source files and compile the system. The aim of this
section is to describe that process in such detail that the presented material is
sufficient to reproduce equivalent or similar results. The guide 
was successfully tested out on the following operating systems: 
Ubuntu 10.04 LTS 64 bit and Ubuntu 12.04 LTS 64 bit. A CD with included source 
code is supplied with the thesis. There is a marking difference between text given in light and
dark grey background color, the former ought to be typed in the terminal window or
it may be an output produced by an application, whereas the later emphasizes a
file modification case. 
\subsection{Installation of OpenBSC}
In order to compile OpenBSC it is required to install the following precompiled
packages\footnote{If more details are required for the installation process a guide can be found at \citep{openbscInstall}.}:

\begin{itemize}\addtolength{\itemsep}{-0.8\baselineskip}
\item libdbi0
\item libdbi0-dev
\item libdbd-sqlite3
\item libortp-dev
\item build-essential
\item libtool
\item autoconf
\item automake
\item git-core
\item pkg-config
\end{itemize}

Before installing the required packages and libraries, to keep the
installation process clean and free of modifying other files, the
author will create a new directory. 
\begin{lstlisting}[backgroundcolor=\color{light-gray}]
mkdir gsm_localization
cd gsm_localization
\end{lstlisting}


By executing the following instructions the required libraries will be installed. 
\begin{lstlisting}[backgroundcolor=\color{light-gray}]
sudo apt-get install libdbi0-dev libdbd-sqlite3 build-essential
sudo apt-get install libtool autoconf automake git-core
sudo apt-get install pkg-config libortp-dev
\end{lstlisting}

After the packages were installed, \textit{libosmocore} library must be downloaded, compiled and installed.
By executing the following instructions: 
\begin{lstlisting}[backgroundcolor=\color{light-gray}][numbers = none]
git clone git://git.osmocom.org/libosmocore.git
cd libosmocore
autoreconf -fi
./configure
make
sudo make install
sudo ldconfig
cd ..
\end{lstlisting}

In the next step \textit{libosmo-abis} will be installed. 
\begin{lstlisting}[backgroundcolor=\color{light-gray}][numbers = none]
git clone git://git.osmocom.org/libosmo-abis.git
cd libosmo-abis
autoreconf -fi
./configure
make
sudo make install
sudo ldconfig
cd ..
\end{lstlisting}

After the previous steps have finished successfully, the author will
proceed with downloading, compiling and installing OpenBSC.
\begin{lstlisting}[backgroundcolor=\color{light-gray}][numbers = none]
git clone git://git.osmocom.org/openbsc.git
cd openbsc/openbsc
autoreconf -i
sudo export PKG_CONFIG_PATH=/usr/local/lib/pkgconfig
./configure
make
\end{lstlisting}

At this point, OpenBSC should be successfully compiled. 
\newpage
\subsection{Configuring nanoBTS for OpenBSC}
To enable the nanoBTS and OpenBSC to be fully operational, the last configuration
steps have to be made. It is necessary to inform the nanoBTS of the IP address of 
the server that is running OpenBSC since it must connect to OpenBSC. We need
to find a free ARFCN channel where our system is
expected to operate\footnote{A licence has to be obtained from the Federal
Network Agency (German: \textit{Bundesnetzagentur}), otherwise it is illegal and may
be considered as a criminal act.}. 

To find the ID and the IP address of the nanoBTS it is required to
start \textit{ipaccess-find}\footnote{The nanoBTS ought to be blinking in orange color
before starting \textit{ipaccess-find}.}.
\begin{lstlisting}[backgroundcolor=\color{light-gray}][numbers = none]
cd ~/gsm_localization/openbsc/openbsc/src/ipaccess
./ipaccess-find
\end{lstlisting}
\textit{ipaccess-find} will produce an output similar to the one given:
\begin{lstlisting}[backgroundcolor=\color{light-gray}][numbers = none]
Trying to find ip.access BTS by broadcast UDP...
MAC_Address='00:02:95:00:61:70'  IP_Address='132.230.4.63'
Unit_ID='1801/0/0'  Location_1=''  Location_2='BTS_NBT131G' 
Equipment_Version='165g029_73' 
Software_Version='168a352_v142b30d0'  
Unit_Name='nbts-00-02-95-00-61-70'  
Serial_Number='00110533'
\end{lstlisting} 
In the next step, the nanoBTS is informed of the OpenBSC IP
address by typing the following commands (the first IP address 
belongs to the server running OpenBSC and the second to the nanoBTS):
\begin{lstlisting}[backgroundcolor=\color{light-gray}][numbers = none]
cd ~/gsm_localization/openbsc/openbsc/src/ipaccess
./ipaccess-config -o 132.230.4.65 132.230.4.63 -r
\end{lstlisting}

It is required to create the directory where the configuration file
will be located and to modify the configuration file.
\begin{lstlisting}[backgroundcolor=\color{light-gray}][numbers = none]
sudo mkdir /usr/local/lcr
cd ~/gsm_localization/openbsc/openbsc/doc/
cd examples/osmo-nitb/nanobts 
sudo cp openbsc.cfg /usr/local/lcr
sudo vim /usr/local/lcr/openbsc.cfg
\end{lstlisting}

A free ARFCN channel can be found using a spectrum analyzer and by
setting the frequency range to the GSM frequency band. One 
has to slide through the frequencies shown on the X-axis,
and by looking at the Y-axis with appropriate frequency 
resolution\footnote{The frequency resolution must be set to 
$f_{CB}=200 \,\mathrm{kHz}$ or higher values for faster movement in the
frequency spectrum.}, where the received power is represented\footnote{
Dependent of the manufacturer and settings of the spectrum analyzer,
it can show signal amplitude, magnitude and power.}. By patiently
observing the Y-axis it can be easily seen on the X-axis which channels are
taken by other GSM service providers and which are free. The chosen channel
ought to be peak free. Once a free frequency channel has been found, it is
necessary to instruct the nanoBTS to operate in that frequency range.
The line, numbered 58, has to be modified with the
correct free ARFCN channel,in this case 877. 
\begin{lstlisting}
arfcn 877
\end{lstlisting}
The ARFCN channel value can be
calculated using the given formula in \eqref{eq:arfcn}, where $f_{start}$
is the starting frequency of the uplink bandwidth for DCS1800, 
$f_{CB}$ is the channel bandwidth and \textit{Offset} is the offset\footnote{
A table with frequency channels can be found at the following URL:
\url{https://gsm.ks.uni-freiburg.de/arfcn.php}}.

\begin{equation}
\label{eq:arfcn}
\centering
\begin{array}{l}
\displaystyle f_{up}(\mathrm{ARFCN}) = f_{start}+f_{CB}\cdot(\mathrm{ARFCN}-\mathrm{Offset}) \\
\displaystyle \\
\displaystyle where \left\{ \begin{array}{rcl}
 f_{start} & = & 1710.2 \,\mathrm{MHz} \\ 
 f_{CB} & = & 200 \,\mathrm{kHz} \\
 \mathrm{Offset} & = & 512
\end{array}\right.
\end{array}
\end{equation}

%Multiple aligned equation
%\begin{equation}
%\label{eq:15}
%\centering
%where \left\{ \begin{array}{rcl}
% f_{start} & = & 1710.2 \,\mathrm{MHz} \\ 
% f_{CB} & = & 200 \,\mathrm{KHz} \\
% \mathrm{Offset} & = & 512
%\end{array}\right.
%\end{equation}


On line numbered 53, the last configuration file modification has to be made for the final configuration of the OpenBSC
software. The Unit ID from the output above has to be set\footnote{Indentation has to match the one of the
configuration file.}. 
\begin{lstlisting}
ip.access unit_id 1801 0
\end{lstlisting}
At this point the nanoBTS and OpenBSC configuration is done. 

\newpage
\subsection{Installation and configuration of RRLP assistance software}
\label{sec:appendSoft}
To install the RRLP software that generates assistance data, several
libraries are required to be installed, \textit{cURL}\footnote{It may happen that
the given download URLs are incorrect and have changed in the meantime, but one
can easily find the latest versions on \url{http://curl.haxx.se/} and \url{http://www.hyperrealm.com/libconfig/}}, 
\textit{libconfig} and \textit{SQLite}. \textit{cURL} was used
for the purpose of safely downloading assistance data from the 
Navigation Center of the US Coast Guard and Trimble server. \textit{libconfig}
library is used for reading in the configuration file, this
way compiling of the software whenever one changes the settings was avoided. The
\textit{SQLite} library was employed to access the database used by OpenBSC to
store the responded data from the mobile stations.
\begin{lstlisting}[backgroundcolor=\color{light-gray}][numbers = none]
cd ~/gsm_localization
sudo apt-get install libsqlite3-dev
wget http://curl.haxx.se/download/curl-7.25.0.tar.gz
wget http://www.hyperrealm.com/libconfig/libconfig-1.4.8.tar.gz
tar -xvzf curl-7.25.0.tar.gz
tar -xvzf libconfig-1.4.8.tar.gz
cd curl-7.25.0
make
sudo make install
cd ..
cd libconfig-1.4.8/
./configure
make
sudo make install
\end{lstlisting}
Once the libraries have been successfully installed, the user may proceed 
with the configuration and compiling the RRLP assistance software, which is the
key software produced in this thesis. The configuration file can be found in the
same directory as the RRLP modules under the name: ``gnssrrlp.cfg''. The sample
configuration file is already preconfigured for the location of ``Angewandte
Mathematik und Rechenzentrum'' building. Latitude and longitude of the BTS are
expressed in decimal degrees and are bounded by \textpm90\textdegree and 
\textpm180\textdegree respectively. Positive latitudes are north of the equator,
whereas negative are south of the equator. It is alike for longitude coordinates, positive
longitudes are east of Prime Meridian and negative are west of the Prime Meridian. 
If the position in decimal degrees of the BTS is unknown, it is straightforward to derive
them using the formula given in \eqref{eq:dd}, where $D$ are degrees, $M$ are 
minutes and $S$ are seconds\footnote{An online converter of the Federal 
Communication Commission can be used as well to convert from degrees, minutes
and seconds to decimal degrees and vice versa
\url{http://transition.fcc.gov/mb/audio/bickel/DDDMMSS-decimal.html}}.
\begin{equation}
\label{eq:dd}
\centering
DD = D + \frac{M}{60} + \frac{S}{3600}
\end{equation}
The altitude may be left as it is, set to 0, since it is not used in the current
measurement technique\footnote{If the value is set to zero,
it is important to set it to 0.0 because \textit{libconfig} would otherwise
convert it to an integer however it is a floating point number.}. The boolean
variables can take \textit{true} or \textit{false} values.
\begin{lstlisting}
// An example configuration file for the GNSS RRLP software.
name = "Configuration for RRLP";
// Change the settings if required:
settings =
{config = ( { 
    ephemeris_url  = "ftp://ftp.trimble.com/pub/eph/CurRnxN.nav";
    almanac_url = "http://www.navcen.uscg.gov/ ?pageName=currentAlmanac&format=yuma";
    latitude_of_BTS  = 48.003601;
    longitude_of_BTS    = 7.848056; 
    altitude_of_BTS  = 0.0; 
    uncertainty_of_lat_long  = 7; 
    uncertainty_of_alt  = 7; 
    confidence_level    = 0; 
    ephemeris_repair  = false; 
    use_reference_time  = false; 
    extra_seconds_to_add  = 7; 
    timezone_of_BTS  = 1; 
    time_to_refresh_ephem = 1;
    time_to_refresh_alm = 1 ; } );};
\end{lstlisting}
The uncertainty of the latitude and longitude correctness can be described
using equation \eqref{eq:unclatlong} \citep{3gppequations}. The uncertainty of
$r$ is expressed in meters, it defines how accurate is the specified location
of the BTS. In the configuration file, $K$ is set to 7, which corresponds to
$r$ = 9.4872 m. 
\begin{equation}
\label{eq:unclatlong}
\centering
\begin{array}{l}
\displaystyle r=C((1+x)^{K}-1)\;\;
\displaystyle where \left\{ \begin{array}{rcl}
 C & = & 10 \\ 
 x & = & 0.1 \\
 K & \in & [0,127] \cap \mathbb{N}_{0}
\end{array}\right.
\end{array}
\end{equation}
A set of uncertainties $r$ is given in table \ref{tab:unclatlong} for various integer values of $K$.
\begin {table}[ht]
\caption{Example uncertainties (latitude and longitude) for various integer values of $K$.}
\label{tab:unclatlong}\centering 
%\rowcolor{2}{light-gray}{}
\scriptsize\fontfamily{iwona}\selectfont
\begin{tabular}{llll}
\toprule 
%$D$&&$P_u$&$\sigma_N$\\
\textbf{Value of $K$}&\textbf{Value of uncertainty $r$}&\textbf{Value of $K$}&\textbf{Value of uncertainty $r$}\\\toprule
0 & 0 m&20 & 57.3 m\\\midrule
1 & 1 m&60 & 3.0348 km
%2 & 2.1 m&100 & 137.8 km\\\midrule
%3 & 3.3 m&- & -
\\\bottomrule
\end {tabular}
\end {table}
Altitude uncertainty can be described using the same Binomial expansion method,
as given in \eqref{eq:uncalt},
however with altered constant values \citep{3gppequations}. The altitude uncertainty ranges between 0 m and 
990.5 m ($h\in[0,990.5]\, \mathrm{m}$). Although the same constant name $K$ is used, it
describes the altitude uncertainty. A set of uncertainties $h$ is given in table \ref{tab:uncalt} for various integer values of $K$.
\begin{equation}
\label{eq:uncalt}
\centering
\begin{array}{l}
\displaystyle h=C((1+x)^{K}-1) \;\;
\displaystyle where \left\{ \begin{array}{rcl}
 C & = & 45 \\ 
 x & = & 0.025 \\
 K & \in & [0,127] \wedge \|K\|
\end{array}\right.
\end{array}
\end{equation}
\begin {table}[]
\caption{Example uncertainties (altitude) for various integer values of $K$.}
\label{tab:uncalt}\centering 
%\rowcolor{2}{light-gray}{}
\scriptsize\fontfamily{iwona}\selectfont
\begin{tabular}{llll}
\toprule 
%$D$&&$P_u$&$\sigma_N$\\
\textbf{Value of $K$}&\textbf{Value of uncertainty $h$}&\textbf{Value of $K$}&\textbf{Value of uncertainty $h$}\\\toprule
0 & 0 m&20 & 28.74 m\\\midrule
1 & 1.13 m&60 & 152.99 m 
%2 & 2.28 m&100 & 486.62 m\\\midrule
%3 & 3.46 m&- & -
\\\bottomrule
\end {tabular}
\end {table}
Confidence level can take any integer value between 0 and 127. The confidence level defines the
percentage of the confidence that the target entity, the GSM user one wants to locate, is within
the geometric shape defined earlier. A value between 0 and 100; 127 may be 
interpreted as ``no information'' \citep{3gppequations}. 
Ephemeris repair is a variable of the boolean type.
Ephemeris data may contain errors or miss some 
satellite information \citep{NASA-Ephem-Errors} \citep{Stanford-Ephem-Errors} 
and the ephemeris repair function, if set to true, will take data of the previous 
measurement report. This introduces an error as well. 
Reference time can be used to provide extra information for the A-GPS in the 
MS of target entity. This field is of boolean type, if set to true, reference
time is included in the sent packets. Since the sent packets are not transmitted 
in real time but put on a stack and then sent to the MS, a time delay exists. 
The reference time being sent to the MS is Coordinated Universal Time (UTC). The GPS device 
receives UTC time from the satellites and adjusts the computer time. To set the correct 
time, time zone offset of the BTS ought to be set correctly.
Finally, the refresh time of downloading new almanac and ephemeris data has to be set.
The variable uses the hour unit, how often the data are being refreshed and downloaded. The almanac
data are valid for up to 180 days \citep{GPS-Guide} but are updated usually every day\footnote{Almanac update times can be found here:
\url{http://www.navcen.uscg.gov/?pageName=currentNanus&format=txt}} \citep{GPS-Pentagon}.

\clearpage
\section{Troubleshooting the BTS}
While the work has been performed on OpenBSC, to open a data channel (SDCCH), 
the BTS was sometimes sent in erroneous states. These states are reported 
through a LED light on the BTS. Based on the color and flash type of the LED 
one can find out the state of the BTS. These states are given in table 
\ref{tbl:LEDStatus} with their appropriate meaning. They may help the 
developer to troubleshoot and find the bug. 

\begin {table}[ht]
\caption{Indicator LED status on the nanoBTS. Table courtesy of \citep{installnanoBTS}.}
\label{tbl:LEDStatus}\centering 
%\rowcolor{2}{light-gray}{}
\scriptsize\fontfamily{iwona}\selectfont
\begin{tabular}{llll}
\toprule 
%$D$&&$P_u$&$\sigma_N$\\
\textbf{State}&\textbf{Color \& Pattern}&\textbf{When}&\textbf{Precedence}\\\toprule
Self-test failure&Red - Steady       &In boot or application code when a power&1 (High) \\
				     &&on self-test fails\\\midrule
Unspecified failure&Red - Steady &On software fatal errors&2\\\midrule
No Ethernet&Orange - Slow flash &Ethernet disconnected&3\\\midrule
Factory reset&Red - Fast blink       &Dongle detected at start up and the&4\\
				     &&factory defaults have been applied\\\midrule
Not configured&Alternating Red/&The unit has not been configured&5\\
	       &Green Fast flash\\\midrule
Downloading code&Orange - Fast flash &Code download procedure is in progress&6\\\midrule
Establishing XML&Orange - Slow blink &A management link has not yet been established&7\\
				      &&but is needed for the TRX to become operational.\\
				      &&Specifically: for a master a Primary OML or\\
				      &&Secondary OML is not yet established; for a\\
				      &&slave an IML to its master or a Secondary \\
				      &&OML is not yet established. \\\midrule
Self-test &Orange - Steady            &From power on until end of backhaul&8\\
				      &&power on self-test\\\midrule
NWL-test 	 &Green - Fast flash& OML established, NWL test in progress&9\\\midrule
OCXO Calibration &Alternating Green/& The unit is in the fast calibrating state [SYNC]&10\\
	         &Orange - Slow blink\\\midrule
Not transmitting &Green - Slow flash & The radio carrier is not being transmitted &11\\\midrule
Operational &Green - Steady & Default condition if none of the above apply&12 (Low)\\\bottomrule
\end {tabular}
\end {table}

\clearpage
\section{Carrier wave demodulation}
\label{sec:carWavDemod}
The reason why the equivalent carrier wave must be generated is straightforward
to understand by looking at the multiplication of two sine waves. 
The GPS L1 signal demodulator at the receiver was depicted in figure
\ref{img:L1Demod}, page \pageref{img:L1Demod}. The incoming signal L1 is multiplied with 
the synthesized sine wave\footnote{Multiplication is the function of
a mixer, denoted as $\otimes$ in figure \ref{img:L1Demod}.}. 
For the purpose of easier analysis and understanding this concept, 
cosine waves shall be used instead of sine waves. The difference between sine
and cosine waves is in the phase shift, as denoted in equation
\eqref{eq:sineEqCosine}. 
\begin{equation}
\label{eq:sineEqCosine}
\sin(\pm x) = \cos\bigg(\frac{\pi}{2} \pm x\bigg)
\end{equation}
Multiplication of two cosine waves, as in equation \eqref{eq:multCosin},
can be derived by adding $\cos(A+B)$ and $\cos(A-B)$ together, as respectively
given in equations \eqref{eq:cos1} and \eqref{eq:cos2}.
\begin{equation}
\label{eq:multCosin}
\cos(A)\cdot\cos(B) = \frac{1}{2}\cos(A-B)+\frac{1}{2}\cos(A+B)
\end{equation}
\begin{equation}
\label{eq:cos1}
\cos(A+B) = \cos(A)\cos(B)-\sin(A)\sin(B)
\end{equation}
\begin{equation}
\label{eq:cos2}
\cos(A-B) = \cos(A)\cos(B)+\sin(A)\sin(B)
\end{equation}
The incoming GPS L1 signal with a frequency $f_{1}$, given in figure \ref{img:L1Demod}, 
can be written as $d_{C/A}\cos(\omega_{1}t)$, a similar form is given in equation \eqref{eq:GPSSignalReceived6},
where $\omega_{1}=2\pi f_{1}$ is
the angle frequency and
$d_{C/A}$ is the C/A data (navigation message modulated with the PRN code),
$d_{C/A}=d_{PRN}\oplus d_{NAV}$.
\begin{equation}
\label{eq:GPSSignalReceived6}
S(t) = \sqrt{\frac{P}{2}}d_{C/A}cos(2\pi f_{c}+\varphi_{GPS}) + n(t)
\end{equation}
If equation \eqref{eq:multCosin} is rewritten with the received GPS signal L1
and synthesized wave with frequency $f_{2}$ substituted, then the equation results the one
given in \eqref{eq:cosResult}
\begin{equation}
\label{eq:cosResult}
d_{C/A}\cdot\cos(\omega_{1}t)\cos(\omega_{2}t) = \frac{1}{2}d_{C/A}\cdot\cos(\omega_{1}t-\omega_{2}t) + \frac{1}{2}d_{C/A}\cos(\omega_{1}t+\omega_{2}t)
\end{equation}
This leaves the resulting signal with two frequency terms, a low frequency 
term $(\omega_{1}t-\omega_{2}t)$
and a high frequency term $(\omega_{1}t+\omega_{2}t)$,
the $t$ can be taken in front of the bracket, as it
is a common multiplier.
The high frequency term, $(\omega_{1}+\omega_{2})$, can be filtered out using
a low-pass filter\footnote{A low-pass filter passes
low frequency signals and attenuates
high frequency signals. In other words, signals higher than the
specified cutoff frequency of the low-pass filter, are cut off by reducing their amplitudes.}.
Ideally, the difference of the angle frequencies is zero,
as in equation \eqref{eq:delaOmega}, since $\cos(\Delta \omega)=\cos(0)=1$
and the remaining left signal is only the C/A code multiplied
with the DC term (zero frequency producing a constant voltage) leaving only $\frac{1}{2}d_{C/A}$.
\begin{equation}
\label{eq:delaOmega}
\Delta \omega = \omega_{1}-\omega_{2} = 0
\end{equation}
However, if the frequencies do not match, $f_{1}\neq f_{2}$,
then the output signal $\frac{1}{2}d_{C/A}$ will be
modified by the residual frequency $f_{1}-f_{2}$, 
and subsequently this will change the demodulated C/A output (also known as phase shift). Under those circumstances
the correlator is unable to match the C/A code with the
correct PRN code. An illustration of this phenomenon is depicted
in figure \ref{img:multCAPhase}. 

\begin{figure}[ht!]
  \centering
  \includegraphics[scale=0.5]{img/PRN-PhaseShiftAfterDemod.pdf}
  \caption{Effects of the low frequency term on the demodulated output
  C/A wave on the GPS receiver (the explanations and figures are from top to bottom).
  If the synthesized frequency is correct, $f_{1}=f_{2}$, the low 
  frequency term becomes a DC term and does not modify the output
  $d_{C/A}$ wave (first figure). If the frequency matches but the
  phase not, in this case the phase is shifted for $\pi$, then
  $d_{C/A}$ is inverted (second figure).
  If the phase shifts with time, then the amplitude and phase of $d_{C/A}$
  will vary as well (third figure). Image courtesy of \citep{diggelen2009a-gps}.}
\label{img:multCAPhase}
\end{figure}

\clearpage
\section{C/A wave demodulation}
\label{sec:CAwaveDemodApend}
The demodulation process, of finding the correct chipping rate,
will examined in this appendix section. 
The chipping period $T_{c}$ can be derived from equation \eqref{eq:chipPeriod}. 
The amount of time required to find a matching PRN code shift, $\tau$,
on the receiver is proportional to the amount of parallely working LFSRs on the system
\citep[Chapter 3]{bensky2008wireless}. Clearly with more LFSRs
the required time for finding the matching phase shift increases.
\begin{equation}
\label{eq:chipPeriod}
T_{c} = \frac{1}{f_{PRN}} = \frac{1}{1.023\cdot 10^6 \mathrm{Hz}}
\end{equation}
To determine whether the synthesized PRN code,
matches the incoming C/A code of the received satellite
signal, known correlation properties of PRN codes are used,
as described in section \ref{sec:gpsDataAndSignal}. 
Since the PRN code is modeled as a sequence of +1's and
-1's, the autocorrelation of
a signal is at its maximum if it is in phase, i.e. 
summing up the sequence products yields the absolute
maximum value for the case when each bit from one signal matches
the bit from the other signal. As an illustration of the idea, an example is
given in figure \ref{img:correlatingSignals}. The cross-correlation
of the incoming C/A code with the first synthesized PRN code produces a 
result of $-3=(+1)\cdot(-1)+(-1)\cdot(+1)+(+1)\cdot(-1)+(+1)\cdot(+1)+(-1)\cdot(+1)$.
However, the cross-correlation of the incoming C/A code
and the second synthesized PRN code yields a result of 
$+5=(+1)\cdot(+1)+(-1)\cdot(-1)+(+1)\cdot(+1)+(+1)\cdot(+1)+(-1)\cdot(-1)$.

\begin{figure}[ht!]
  \centering
  \includegraphics[scale=0.50]{img/Correlation.pdf}
  \caption{Cross-correlation on three different signals. Image courtesy of \citep{understandGPS}.}
\label{img:correlatingSignals}
\end{figure}
The same principle applies to the transmitted C/A and 
generated PRN code sequences in the GPS receiver. Thus, this can be modeled using
the equation given in \eqref{eq:autocorrelationProperty}, 
where $G_{i}(t)$ is the C/A code\footnote{PRN generated codes for GPS satellites 
are called Gold code sequences since they were first discovered by Dr. Robert Gold.} as a
function of time $t$, for the GPS satellite $i$; $T_{C/A}$ is the
C/A chipping period of $977.5 \,ns$ and $\tau$ is the phase shift 
in the auto-correlation function \citep[Chapter 4]{understandGPS}.

\begin{equation}
\label{eq:autocorrelationProperty}
R_{i}(t) = \frac{1}{1023\cdot T_{C/A}} \int_{t=0}^{1022} G_{i}(t)G_{i}(t+\tau)d\tau
\end{equation}
Another correlation property of the PRN codes is used,
the fact that in the ideal case the cross-correlation of two
different PRN codes yields a result of zero. The ideal case of
PRN code can be modeled as in equation \eqref{eq:prnIdealCaseZero}.
\begin{equation}
\label{eq:prnIdealCaseZero}
R_{ij}(\tau) = \int_{-\infty}^{+\infty} PRN_{i}(t)PRN_{j}(t+\tau)d\tau = 0
\end{equation}
$PRN_{i}$ is the PRN code waveform for GPS satellite $i$ and 
$PRN_{j}$ is the PRN code waveform for every other GPS satellite other
than $i$, $i\neq j$ \citep[Chapter 4]{understandGPS}. Equation
\eqref{eq:prnIdealCaseZero} ``states that the PRN waveform of satellite
$i$ does not correlate with PRN waveform of any other satellite $j$ for
any phase shift $\tau$'' \citep[Chapter 4]{understandGPS}.
Without the property given in \eqref{eq:prnIdealCaseZero},
the GPS receiver would not be able to smoothly 
differentiate between different GPS satellite signals. 
Once the phase shift, $\tau$, has been found, the C/A code is modulated
(XORed) with it. The resulting binary code are the transmitted subframes containing data
required to estimate the position.
%The implementation problem of finding correct C/A and carrier wave demodulation shall be
%further explained in the following section \ref{sec:2dSearch}.

\clearpage
\section{GPS assistance data descriptions}
Description of assistance data that are converted and sent inside the RRLP protocol.
\begin {table}[ht!]
\caption{Almanac message. Table courtesy of \citep{harper2010server-side}.}
\label{tbl:almanacMessage}\centering 
%\rowcolor{2}{light-gray}{}
\scriptsize\fontfamily{iwona}\selectfont
\begin{tabular}{lll}
\toprule 
%$D$&&$P_u$&$\sigma_N$\\
\textbf{Field (IE)}& \textbf{Description}\\\toprule
SatelliteID&This is the satellite ID that is in the range of 0 to 63. PRN=SatelliteID + 1\\\midrule
SV Health&Satellite health (e.q. 000 means the satellite is fully operational)\\\midrule
$e$&``Eccentricity shows the amount of the orbit deviation from circular (orbit). It is the distance\\
&between the foci divided by the length of the semi-major axis'' \citep{ubxGPSDict}\\\midrule
TOA&Time of applicability, reference time for orbit and clock parameters (seconds). ``The number of\\
&seconds in the orbit when the almanac data were generated'' \citep{ubxGPSDict}\\\midrule
OI&Orbital inclination (radians). The angle to which the SV orbit meets the equator \citep{ubxGPSDict}\\\midrule
RORA&Rate or right ascension (radians/second). ``Rate of change of the angle of right ascension as\\
&defined in the Right Ascension mnemonic'' \citep{ubxGPSDict}\\\midrule
$A^{1/2}$& Square root of semi-major axis (meters$^{1/2}$). `` This is defined as the measurement from the center\\
&of the orbit to either the point of apogee or the point of perigee'' \citep{ubxGPSDict}\\\midrule
$\Omega_0$& Right Ascension at Week (radians). Longitude of ascending node of orbit plane at weekly epoch\\\midrule
$\omega$&Argument of perigee (semicircles).  ``An angular measurement along the orbital path measured from\\
&the ascending node to the point of perigee, measured in the direction of the SV's motion'' \citep{ubxGPSDict}\\\midrule
$M_0$&Mean anomaly (radians)\\\midrule
$a_{f0}$&Satellite clock bias (seconds). Satellite clock error at reference time\\\midrule
$a_{f1}$&Satellite clock drift (seconds per second). Satellite clock error rate\\\midrule
Week&Week number since the last reset (i.e. since year 1980 modulo 1024 weeks)
\\\bottomrule
\end {tabular}
\end {table}

\begin{table}[hc]
\scriptsize\fontfamily{iwona}\selectfont
\begin{minipage}[b]{.49\textwidth}
  \centering
  \begin{tabular}{ll}
\toprule 
%$D$&&$P_u$&$\sigma_N$\\
\textbf{Field (IE)} & \textbf{Description}\\\toprule
$\alpha_{0}$&Coefficient 0 of vertical delay\\\midrule
$\alpha_{1}$&Coefficient 1 of vertical delay\\\midrule
$\alpha_{2}$&Coefficient 2 of vertical delay\\\midrule
$\alpha_{3}$&Coefficient 3 of vertical delay\\\midrule
$\beta_{0}$&Coefficient 0 of period of the model\\\midrule
$\beta_{1}$&Coefficient 1 of period of the model\\\midrule
$\beta_{2}$&Coefficient 2 of period of the model\\\midrule
$\beta_{3}$&Coefficient 3 of period of the model
\\\bottomrule
\end {tabular}
  \caption{GPS Ionosphere Model.}
  \label{tbl:ionoModel}
\end{minipage}
\begin{minipage}[b]{.43\textwidth}
  \centering
\begin{tabular}{ll}
\toprule 
%$D$&&$P_u$&$\sigma_N$\\
\textbf{Field (IE)} & \textbf{Description}\\\toprule
$A_{1}$&Drift coefficient of GPS time scale relative\\
&to UTC time scale\\\midrule
$A_{0}$&Bias coefficient of GPS time scale relative\\
&to UTC time scale\\\midrule
$t_{ot}$&Time data reference time of week\\\midrule
$\Delta t_{LS}$&Current or past leap second count\\\midrule
$WN_{0}$&Time data reference week number\\\midrule
$WN_{LSF}$&Leap second reference week number\\\midrule
$DN$&Leap second reference day number\\\midrule
$\Delta t_{LSF}$&Current of future leap second count
\\\bottomrule
\end {tabular}
  \caption{GPS UTC Model.}
  \label{tbl:utcModel}
\end{minipage}
\end{table}

\newpage
\begin {table}[ht!]
\caption{Navigation message (ephemeris). Table courtesy of \citep{harper2010server-side}.}
\label{tbl:navMessage}\centering 
%\rowcolor{2}{light-gray}{}
\scriptsize\fontfamily{iwona}\selectfont
\begin{tabular}{llll}
\toprule 
%$D$&&$P_u$&$\sigma_N$\\
\textbf{Field (IE)} & \textbf{Description}\\\toprule
Satellite ID&This is the satellite ID that is in the range of 0 to 63. PRN=SatelliteID + 1\\\midrule
Satellite status&This is an indicator of whether this is a new or existing satellite and whether\\
&the navigation model is new or the same.\\\midrule
C/A or P on L2&Code(s) on L2 channel\\\midrule
URA Index&User range accuracy\\\midrule
SV Health&Satellite health\\\midrule
IODC&Issue of data, clock\\\midrule
L2 P Data flag& \\\midrule
SF 1 Reserved& \\\midrule
$T_{GD}$&Estimated group delay differential\\\midrule
$t_{oc}$&Apparent clock correction\\\midrule
$a_{f2}$&Apparent clock correction\\\midrule
$a_{f1}$&Apparent clock correction\\\midrule
$a_{f0}$&Apparent clock correction\\\midrule
$C_{rs}$&Amplitude of the sine harmonic correction term to the orbit radius (meters)\\\midrule
$\Delta n$&Mean motion difference from computed value (semicircles/second)\\\midrule
$M_{0}$&Mean anomaly at reference time (semicircles)\\\midrule
$C_{uc}$&Amplitude of the cosine harmonic correction term to the\\
&argument of latitude (radians)\\\midrule
$e$&Eccentricity\\\midrule
$C_{us}$&Amplitude of the sine harmonic correction term to the argument of latitude\\
&(radians)\\\midrule
$A^{1/2}$&Square root of semi-major axis (meters)\\\midrule
$t_{oe}$&Reference time ephemeris\\\midrule
Fit Interval Flag&\\\midrule
AODO&Age of data offset\\\midrule
$C_{ic}$&Amplitude of the cosine harmonic correction term to the angle of inclination\\
&(radians)\\\midrule
$\Omega_0$&Longitude of ascending node of orbit plane at weekly epoch (semicircles)\\\midrule
$C_{is}$&Amplitude of the cosine harmonic correction term to the angle of inclination\\
&(radians)\\\midrule
$i_{0}$&Inclination angle at reference time (semicircles)\\\midrule
$C_{rc}$&Amplitude of the cosine harmonic correction term to the orbit radius (meters)\\\midrule
$\omega$&Argument of perigee (semicircles)\\\midrule
OMEGAdot&Rate of right ascension (semicircles/second)\\\midrule
Idot&Rate of inclination angle (semicircles/second)
\\\bottomrule
\end {tabular}
\end {table}
\newpage



\clearpage
\section{GPS distance and position estimation}
\label{sec:distanceAndPosition}
In this appendix section the focus is set on distance and position estimation inside of the GPS receiver.
GPS system, as discussed earlier, takes advantage of the TOA ranging concept
to determine user's position. It is measured how long it takes for a signal to arrive from a
known location to the current unknown position on Earth.
\begin{figure}[ht!]
  \centering
  \includegraphics[scale=0.50]{img/Localization.pdf}
  \caption{Basic distance estimation principle for one satellite. Image courtesy of \citep{understandGPS}.}
\label{img:SatLocalization}
\end{figure}
In figure \ref{img:SatLocalization}, an example concept can be seen, where $\vec{u}=(x_u,y_u,z_u)$ represents the unknown 
GPS user position vector with respect to Earth-Centered, Earth-Fixed\footnote{ECEF is a Cartesian coordinate system
where the point $(0,0,0)$ is defined as the center of mass of the Earth \citep{earthCoordinates}.} 
(ECEF) coordinate system,  $\vec{r}$ is the distance vector from the satellite to the user and  $\vec{s}=(x_s,y_s,z_s)$
represents the GPS satellite position with respect to ECEF at a time-point $t_1$. 
$t_1$ is the time-point when the time stamp was generated on the GPS satellite.
Vector $\vec{s}$ is computed from ephemeris data broadcasted
by the satellite. The distance vector $\vec{r}$, which is the distance between the satellite and the GPS receiver, can be computed using equation \eqref{eq:r}
and its magnitude is given in equation \eqref{eq:rMag}.
\begin{equation}
\label{eq:r}
\vec{r}=\vec{s}-\vec{u}
\end{equation}
\begin{equation}
\label{eq:rMag}
r=\Vert s-u\Vert
\end{equation}
The geometric distance of $r$ is computed by measuring the signal propagation time, this is illustrated in figure \ref{img:TimingLoc}. 
The PRN code generated on the GPS satellite 
at time $t_1$ arrives at the time $t_2$. The difference between these two time stamps, $\Delta t$, represents the 
propagation time. By multiplying the propagation time, $\Delta t$, with the speed of light, $c$, the 
geometric distance $r$ is computed, as given in equation \eqref{eq:rDist}. 
\begin{figure}[ht!]
  \centering
  \includegraphics[scale=0.50]{img/TimingLoc.pdf}
  \caption{Estimating the distance by phase shift $\Delta t =t_2 - t_1 =\tau$. Image courtesy of \citep{understandGPS}.}
\label{img:TimingLoc}
\end{figure}
\begin{equation}
\label{eq:rDist}
r=c\Delta t
\end{equation}
Since the clocks are not synchronized, clock error offsets have to be added to 
the geometric distance $r$. This new distance is called \textit{pseudorange}\footnote{pseudo - Not genuine; sham; not perfect.}, $\rho$, because the range is 
determined using the difference of two nonsynchronized clocks that generate PRN codes (one on the GPS satellite and
the other one on the receiver). Pseudorange is calculated as given in equation \eqref{eq:rho}, where 
$t_{u}$ is the advance of the receiver clock with respect to the system time\footnote{$t_{u}$ or system
time is the exact time on Earth and it is the most precise time known!} and $\delta t$ is the offset of the
satellite clock from the system time \citep{understandGPS}.
\begin{equation}
\label{eq:rho}
\rho=r + c(t_{u}-\delta t)
\end{equation}
Equation \eqref{eq:rMag} can be rewritten as \eqref{eq:rhoR} with respect to equation \eqref{eq:rho}.
\begin{equation}
\label{eq:rhoR}
\rho - c(t_{u}-\delta t) = \Vert s-u\Vert
\end{equation}
Offset of the satellite clock from the system time, $\delta t$, is updated from Earth and it is inside
the GPS transmitted data. For that reason, it can be removed for sake of simplicity, i.e. it is not an unknown term anymore,
then the equation \eqref{eq:rhoR} can be rewritten as \eqref{eq:rhoNew}.
\begin{equation}
\label{eq:rhoNew}
\rho - ct_{u} =  \Vert s-u\Vert
\end{equation}
In order to estimate user's (GPS receiver) position,
advance of the receiver clock with respect to the system time, $t_u$, has to be found, in other words equation \eqref{eq:rhoSats}
has to be solved, where $i$ is the index of visible satellites at the moment of signal reception \citep{understandGPS}.
\begin{equation}
\label{eq:rhoSats}
\rho_i= \Vert s_i-u\Vert + ct_u
\end{equation}
The estimated position of the user, $\vec{u}=(x_u,y_u,z_u)$, is a three dimensional vector and as stated
above the clock offset, $t_u$, is unknown as well. This four dimensional space requires to have at least four pseudorange
equations \eqref{eq:rhoSats} to find all the four unknown terms. 
As a consequence of this fact, at least four satellites have to be visible at 
the same time to estimate user's position. Equation given in \eqref{eq:rhoSats} take the form in
\eqref{eq:rhoSatsNew} because the coordinate system is Cartesian and  $\rho_i$ is nothing else but Euclidean distance
where $i=1,2,...,n$ such that $n\geq4$, and $\vec{s_i}=(x_i,y_i,z_i)$ is the satellite position estimated from the ephemeris data.
\begin{equation}
\label{eq:rhoSatsNew}
\rho_i= \sqrt{(x_i-x_u)^2+(y_i-y_u)^2+(z_i-z_u)^2} + ct_u
\end{equation}
Undoubtedly, the given equation in \eqref{eq:rhoSatsNew} is a nonlinear 
equation\footnote{Nonlinear
equations, also known as polynomial equations, are equations that can not satisfy both
of the linearity properties:
additivity $f(x+y)=f(x)+f(y)$ and homogeneity $f(\alpha x) = \alpha f(x)$, $\alpha \in \mathbb{R}$ \citep{nonlinear}.}.
It is not straightforward to find explicit solutions of nonlinear equations. It is more difficult to find the solution 
compared to linear equations.
There are different techniques to solve sets of nonlinear equations \citep[Chapter 7]{understandGPS}
but in this work the linearisation method\footnote{Linear approximation is a technique where a function
is approximated using a linear function.}
shall be presented to find the unknown terms $(x_u,y_u,z_u,t_u)$. In other words, out of an approximate position 
and clock offset, the true clock offset will be calculated. Out of this calculation will follow the true user position.

Let the equation \eqref{eq:rhoSatsNew} for pseudoranges, be rewritten as a function $f$ of four
unknown terms $x_u$, $y_u$, $z_u$ and $t_u$, as
given in \eqref{eq:rhoSatsNewFun} \citep[Chapter 2]{understandGPS}. 
\begin{equation}
\label{eq:rhoSatsNewFun}
\rho_i= \sqrt{(x_i-x_u)^2+(y_i-y_u)^2+(z_i-z_u)^2} + ct_u = f(x_u,y_u,z_u,t_u)
\end{equation}
Suppose that the approximation of the
position and the clock offset are known,
denoted as $\hat{x_u}$, $\hat{y_u}$, $\hat{z_u}$ and $\hat{t_u}$, then equation \eqref{eq:rhoSatsNewFun}
can be rewritten as an approximate pseudorange \eqref{eq:rhoSatsNewFunApprox}.
\begin{equation}
\label{eq:rhoSatsNewFunApprox}
\hat{\rho_i}= \sqrt{(x_i-\hat{x_u})^2+(y_i-\hat{y_u})^2+(z_i-\hat{z_u})^2} + c\hat{t_u} = f(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u})
\end{equation}
In other words, the unknown true position terms $x_u$, $y_u$, $z_u$ and the clock offset term $t_u$, of the GPS receiver,
are expressed by the approximate values and an incremental component as shown in equation \eqref{eq:userCoordinates} \citep{understandGPS}.
\begin{equation}
\label{eq:userCoordinates}
\begin{array}{l}
x_u = \hat{x_u}+\Delta x_u \\
y_u = \hat{y_u}+\Delta y_u \\
z_u = \hat{z_u}+\Delta z_u \\
t_u = \hat{t_u}+\Delta t_u
\end{array} 
\end{equation}
By inserting the terms from \eqref{eq:userCoordinates} into equation \eqref{eq:rhoSatsNewFun}, a new equation is derived
as in \eqref{eq:rhoSatsNewFunwithApprox}.
\begin{equation}
\label{eq:rhoSatsNewFunwithApprox}
f(x_u,y_u,z_u,t_u) = f(\hat{x_u}+\Delta x_u, \hat{y_u}+\Delta y_u, \hat{z_u}+\Delta z_,\hat{t_u}+\Delta t_u)
\end{equation}
In the next step the pseudorange function shall be approximated using Taylor series (linearisation of the nonlinear equation)\footnote{Taylor
series ``is a representation of a
function as an infinite sum of terms that are calculated from the values of the function's
derivatives at a single point'' \citep[Chapter 11]{taylor}.}. Taylor
series for a function $f(x)$ is given in equation \eqref{eq:taylor}, where as $a$ approaches $x$ the estimation
error shall be smaller and smaller, i.e. $f(x) = f(a)$ when $x=a$. The approximation error
depends on Taylor polynomial degree (the amount of terms or taken derivatives of the function)
and how far away the point $a$ is from $x$ \citep[Chapter 11.9]{taylor}.
The basic idea of the principle can be seen in figure \ref{img:taylorSeries}.
\begin{equation}
\label{eq:taylor}
f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + \frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+...
\end{equation}

\begin{figure}[ht!]
  \centering
  \includegraphics[scale=0.50]{img/TaylorSeries.pdf}
  \caption{Taylor series approximation for a point $a=0.5$ where $n$ is the Taylor polynomial degree.}
\label{img:taylorSeries}
\end{figure}
Due to the four unknown terms, Taylor series for multi-variables
have to be used. The general formula is given in
equation \eqref{eq:Multitaylor}, where vector $\mathbf{x}\in\mathbb{R}^n$ denotes
$n$ variables, $\nabla$ (nabla) is the Del\footnote{Del, $\nabla$,
is the vector differential operator.} operator given in \eqref{eq:Del} and $\mathbf{a}$ is the 
linearisation point of interest
\citep{multiTaylor}.
\begin{equation}
\label{eq:Multitaylor}
f(\mathbf{x}) \approx f(\mathbf{a}) + \nabla f |_{\mathbf{x=a}} \cdot (x-a)
\end{equation}
\begin{equation}
\label{eq:Del}
\nabla^{T} = \left[\frac{\partial}{\partial x_{1}}...\frac{\partial}{\partial x_{n}}\right]
\end{equation}
One can note that in equation \eqref{eq:Multitaylor} the Taylor series polynomial is of the first degree.
This is because of one reason, it linearizes the approximation of the function $f(\mathbf{x})$ at point $\mathbf{a}$
and as a consequence it removes the nonlinearities \citep{understandGPS} \citep[Chapter 11.10]{taylor}, as seen
in figure \ref{img:taylorSeries}, for $n=1$ the resulting function is linear.
In the previously described step, one would calculate a hyperplane tangent to a
point $a$ in a $n$-Dimensional space. By inserting equation \eqref{eq:rhoSatsNewFunwithApprox} in 
equation \eqref{eq:Multitaylor}, it yields equation \eqref{eq:MultitaylorFour} where $\mathbf{x}=(x_u,y_u,z_u,t_u)$
and $\mathbf{a}=(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u})$.
\begin{equation}
\label{eq:MultitaylorFour}
\begin{array}{l}
f(\hat{x_u}+\Delta x_u, \hat{y_u}+\Delta y_u, \hat{z_u}+\Delta z_,\hat{t_u}+\Delta t_u) \approx
		     f(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u}) \\[0.5em]
		    + \dfrac{\partial f(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u})}{\partial \hat{x_u}}\Delta x_u 
+\dfrac{\partial f(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u})}{\partial \hat{y_u}}\Delta y_u \\
+\dfrac{\partial f(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u})}{\partial \hat{z_u}}\Delta z_u 
+\dfrac{\partial f(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u})}{\partial \hat{t_u}}\Delta t_u 
\end{array}
\end{equation}
The terms from equation \eqref{eq:MultitaylorFour} are solved individually in
equations \eqref{eq:MultitaylorDeriv} where $\sqrt{(x_i-\hat{x_u})^2+(y_i-\hat{y_u})^2+(z_i-\hat{z_u})^2}$
has been substituted with $\hat{r_i}$.
\begin{equation}
\label{eq:MultitaylorDeriv}
\begin{array}{l}
\dfrac{\partial f(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u})}{\partial \hat{x_u}} = \dfrac{1}{2}\dfrac{-2(x_{i}-\hat{x_{u}})}{\sqrt{(x_i-\hat{x_u})^2+(y_i-\hat{y_u})^2+(z_i-\hat{z_u})^2}}
=-\dfrac{x_i-\hat{x_u}}{\hat{r_i}}\\[0.9em]
\dfrac{\partial f(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u})}{\partial \hat{y_u}} = \dfrac{1}{2}\dfrac{-2(y_{i}-\hat{y_{u}})}{\sqrt{(x_i-\hat{x_u})^2+(y_i-\hat{y_u})^2+(z_i-\hat{z_u})^2}}
=-\dfrac{y_i-\hat{y_u}}{\hat{r_i}}\\[0.9em]
\dfrac{\partial f(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u})}{\partial \hat{z_u}} = \dfrac{1}{2}\dfrac{-2(z_{i}-\hat{z_{u}})}{\sqrt{(x_i-\hat{x_u})^2+(y_i-\hat{y_u})^2+(z_i-\hat{z_u})^2}}
=-\dfrac{z_i-\hat{z_u}}{\hat{r_i}}\\[0.9em]
\dfrac{\partial f(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u})}{\partial \hat{t_u}} = c
\end{array}
\end{equation}
This is followed by substituting the equation terms from \eqref{eq:MultitaylorDeriv}, \eqref{eq:rhoSatsNewFun} and \eqref{eq:rhoSatsNewFunApprox}
into \eqref{eq:MultitaylorFour}, the resulting equation is given in \eqref{eq:MultitaylorDerivAfter}.
\begin{equation}
\label{eq:MultitaylorDerivAfter}
\begin{array}{l}
\rho_i = \hat{\rho_i} -\dfrac{x_i-\hat{x_u}}{\hat{r_i}}\Delta x_u -\dfrac{y_i-\hat{y_u}}{\hat{r_i}}\Delta y_u -\dfrac{z_i-\hat{z_u}}{\hat{r_i}}\Delta z_u + c\Delta t_u
\end{array}
\end{equation}
At this step, by solving equation \eqref{eq:MultitaylorFour}, the linearisation of the nonlinear equations is completed. 
\begin{equation}
\label{eq:MultitaylorDerivAfterRearange}
\begin{array}{l}
\hat{\rho_i} - \rho_i = \dfrac{x_i-\hat{x_u}}{\hat{r_i}}\Delta x_u +\dfrac{y_i-\hat{y_u}}{\hat{r_i}}\Delta y_u +\dfrac{z_i-\hat{z_u}}{\hat{r_i}}\Delta z_u - c\Delta t_u
\end{array}
\end{equation}
\begin{equation}
\label{eq:SubsTerms1}
\Delta\rho = \hat{\rho_i} - \rho_i \\[0.7em]
\end{equation}
\begin{equation}
\label{eq:SubsTerms2}
\alpha_{xi} = \dfrac{x_i - \hat{x_u}}{\hat{r_i}} \hspace{1.5em} \alpha_{yi} = \dfrac{y_i - \hat{y_u}}{\hat{r_i}} \hspace{1.5em} \alpha_{zi} = \dfrac{z_i - \hat{z_u}}{\hat{r_i}}
\end{equation}
By rearranging the equation \eqref{eq:MultitaylorDerivAfter} one derives equation \eqref{eq:MultitaylorDerivAfterRearange}.
And then by substituting the terms in \eqref{eq:SubsTerms1} and \eqref{eq:SubsTerms2} into \eqref{eq:MultitaylorDerivAfterRearange}, 
the equation resembles the equation in \eqref{eq:userPosition}.
\begin{equation}
\label{eq:userPosition}
\Delta\rho_i = \alpha_{xi}\Delta x_u + \alpha_{yi}\Delta y_u + \alpha_{zi}\Delta z_u - c\Delta t_u
\end{equation}
There are four unknowns, $\Delta x_u$, $\Delta y_u$, $\Delta z_u$ and $\Delta t_u$, in equation \eqref{eq:userPosition}.
By solving this set of linear equations, which shall result in finding of $\Delta x_u$, $\Delta y_u$, $\Delta z_u$ and $\Delta t_u$,
the GPS receiver position is computed. The GPS receiver position $(x_u, y_u, z_u)$ and clock offset $t_u$ are obtained by substituting 
them into equations in \eqref{eq:userCoordinates}. Equation \eqref{eq:userPosition} can be rewritten for four satellites 
in the matrix form as in \eqref{eq:userPositionMatrix}.
\begin{equation}
\label{eq:userPositionMatrix}
\Delta\boldsymbol{\rho} = \boldsymbol{\alpha} \Delta \boldsymbol{x}
\end{equation}

\begin{equation}
\Delta\boldsymbol{\rho}=
\begin{bmatrix}
\Delta \rho_1 \\
\Delta \rho_2 \\
\Delta \rho_3 \\
\Delta \rho_4 
\end{bmatrix}
\hspace{1.5em}
\boldsymbol{\alpha}=
\begin{bmatrix}
\alpha_{x1} & \alpha_{y1} & \alpha_{z1} & 1 \\
\alpha_{x2} & \alpha_{y2} & \alpha_{z2} & 1 \\
\alpha_{x3} & \alpha_{y3} & \alpha_{z3} & 1 \\
\alpha_{x4} & \alpha_{y4} & \alpha_{z4} & 1 
\end{bmatrix}
\hspace{1.5em}
\Delta \boldsymbol{x}=
\begin{bmatrix} 
\Delta x_u \\
\Delta y_u \\
\Delta z_u \\
-\Delta ct_u 
\end{bmatrix}
\end{equation}
Finally, by multiplying both left sides\footnote{Matrix multiplication is not commutative, $\mathbf{AB\neq BA}$.} 
of the equation \eqref{eq:userPositionMatrix} with the inverse term of $\boldsymbol{\alpha}$, it yields the result 
of the unknown terms, as given in equation \eqref{eq:userPositionMatrixFinal}.
\begin{equation}
\label{eq:userPositionMatrixInverseMult}
\boldsymbol{\alpha}^{-1}\Delta\boldsymbol{\rho} = \boldsymbol{\alpha}^{-1}\boldsymbol{\alpha} \Delta \boldsymbol{x}
\end{equation}
\begin{equation}
\label{eq:userPositionMatrixFinal}
\Delta \boldsymbol{x} = \boldsymbol{\alpha}^{-1} \Delta\boldsymbol{\rho}
\end{equation}
Linearisation is repeated in a loop, where in the next round the approximate positions are set
to the just derived position values, that is, $\hat{x_u}=x_u$, $\hat{y_u}=y_u$, $\hat{z_u}=z_u$ and
$\hat{t_u}=t_u$. This process is repeated until the approximated positions converge to their final
values. It is not necessarily required that the initial positions are very accurate
and the results are usually obtained by 4-5 iterations \citep{pseudorangeError}. 
Risks exist that the solution may be still be corrupted but there are different error avoiding 
mechanisms to solve these problems, like minimizing the error contribution using more than four satellite
measurements \citep{pseudorangeError} \citep[Chapter 7]{understandGPS}. 

%\clearpage
%\section{GPS Constants and equations}
%\label{sec:gpsConsAndEq}
%\begin{alignat}{4}
% & A & = & \; (\sqrt{A})^2 \nonumber \\
% & n_{0} & = &\; \sqrt{\frac{\mu}{A^3}} \nonumber \\
% & t_{k} & = &\; t-t_{oe} \nonumber \\
% & n & = &\; n_{0} + \Delta n \nonumber \\
% & M_{k} & = &\; M_{0} + nt_{k} \nonumber \\
% & M_{k} & = &\; E_{k} - e\sin E_{k} \nonumber \\
% & v_{k} & = & \tan ^{-1} \left( \frac{\sin v_{k}}{\cos v_{k}} \right) = \tan ^{-1} \left( \frac{\frac{\sqrt{1-e^2} \sin E_{k}}{1-e \cos E_{k}}}{\frac{\cos E_{k}-e}{1-e\cos E_{k}}} \right) \nonumber \\
% & v_{k} & = & \tan ^{-1} \left( \frac{\sin v_{k}}{\cos v_{k}} \right) = \tan ^{-1} \left( \frac{\sqrt{1-e^2} \sin E_{k}/(1-e \cos E_{k})}{(\cos E_{k}-e)/(1-e\cos E_{k})} \right) = \tan ^{-1} \left( \frac{\sqrt{1-e^2} \sin E_{k}}{\cos E_{k} - e} \right) \nonumber \\
% & E_{k} & = & \cos ^{-1} \left( \frac{e+\cos v_{k}}{1+e \cos v_{k}} \right) \nonumber \\
% & \Phi_{k} & = &\; v_{k} + \omega \nonumber \\
% & \delta u_{k} & = &\;  c_{us} \sin{2\Phi_{k}} + C_{us} \cos{2\Phi_{k}}  \\
% & \delta r_{k} & = &\;  c_{rc} \cos{2\Phi_{k}} + C_{rs} \sin{2\Phi_{k}} \nonumber \\
% & \delta i_{k} & = &\;  c_{ic} \cos{2\Phi_{k}} + C_{is} \sin{2\Phi_{k}} \nonumber \\
% & u_{k} & = &\;  \Phi_{k} + \delta u_{k} \nonumber \\
% & r_{k} & = &\; A(1-e\cos{E_{k}})+\delta r_{k} \nonumber \\
% & i_{k} & = &\; i_{0} + \delta i_{k} + (IDOT)t_{k} \nonumber \\
% & x_{k}^{'} & = &\; r_{k} \cos{u_{k}} \nonumber \\
% & y_{k}^{'} & = &\; r_{k} \sin{u_{k}} \nonumber \\
% & \Omega_{k} & = &\; \Omega_{0} + (\Omega - \Omega_{e})t_{k} - \Omega_{e}t_{oe} \nonumber \\
% & x & = &\; x_{k}^{'} \cos{\Omega_{k}}-y_{k}^{'}\cos{i_{k}}\sin{\Omega_{k}} \nonumber \\
% & y & = &\; x_{k}^{'} \sin{\Omega_{k}}-y_{k}^{'}\cos{i_{k}}\cos{\Omega_{k}} \nonumber \\
% & z & = &\; y_{k}^{'} \sin{i_{k}} \nonumber
%\end{alignat}

%\begin{equation}
%\label{eq:paramconst1}
%  \begin{split}
%  \mu_{e} = 3.986004418\cdot 10^{14} \frac{m^3}{s^2}
%  \end{split}
%\quad\Longleftarrow\quad
%  \begin{split}
%    \mbox{Geocentric gravitational constant}
%  \end{split}
%\end{equation}


%\begin{equation}
%\label{eq:paramconst2}
%  \begin{split}
%  c= 2.99792458\cdot 10^{8} \frac{m}{s}
%  \end{split}
%\quad\Longleftarrow\quad
%  \begin{split}
%    \mbox{speed of light}
%  \end{split}
%\end{equation}