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diff --git a/vorlagen/thesis/src/kapitel_x.tex b/vorlagen/thesis/src/kapitel_x.tex
index c81182c..0a1441a 100644
--- a/vorlagen/thesis/src/kapitel_x.tex
+++ b/vorlagen/thesis/src/kapitel_x.tex
@@ -19,44 +19,47 @@ inside the GSM network
\label{img:gpsprinciple}
\end{figure}
+
+
The GPS satellites\footnote{Satellites are named as space vehicles
and the abrevation SV is used in the equation notations
to denote a parameter related to the satellite itself.}
-orbiting our planet have precise clocks on board.
+orbiting our planet, at a distance of approximately 20,200 km,
+are equiped with precise atomic clocks \citep[Chapter 2.7]{diggelen2009a-gps}.
These atomic clocks are calibrated and maintained on
a daily basis by the U.S. Air Force, \citep{GPS-Pentagon}.
-The time the clocks generate is called \textit{GPS
-system time}, $t_{SV}$, and it is generated at the moment
-of the frame broadcast \citep{GPS-Interface-Specification}.
-In order to calculate the signal travel time,
-between the satellite and the GPS receiver, their precise
-time difference needs to be known. Therefore, each
-satellite signs the frame before its sent with its exact
-broadcast time. The broadcast time is located in the
+The time the clock generates is called \textit{GPS
+system time}, denoted as $t_{SV}$,
+and it is generated as a time stamp at the moment
+of the frame broadcast \citep{GPS-Interface-Specification}.
+Each satellite signs the frame with its exact
+broadcast time. The broadcast time is encapsulated in the
subframe 1 of the 1500 bit long frame. In addition to the
broadcast time, subframe 1 contains parameters to account
-for the deterministic clock errors emodied in the
-broadcast GPS system time stamp. These errors can be
+for the deterministic clock errors embedded in the
+broadcasted GPS system time stamp. These errors can be
characterized as bias, drift and aging errors
\citep{GPS-Interface-Specification}. The correct broadcast
-time can be estimated using the model equation given in
+time, denoted as $t$, can be estimated using the model equation given in
\eqref{eq:timecorrection1} \citep{GPS-Interface-Specification}.
-$t$ is the correctly estimated GPS system time at broadcast
-moment. In equation \eqref{eq:timecorrection2}, where the GPS
+In equation \eqref{eq:timecorrection2}, where the GPS
receiver is required to calculate the satellite clock
-offset, $\Delta t_{SV}$, a number of unknown terms can be
-seen. These terms are contained in the subframe 1 or
+offset, denoted as $\Delta t_{SV}$, a number of unknown terms can be
+seen. These terms are encapsulated in the subframe 1 or they
can be estimated using predefined equations. The polynomial
-coefficients: $a_{f0}$ - clock offset, $a_{f1}$ - fractional
-frequency offset, $a_{f0}$ - fractional frequency drift; and
-$t_{0c}$ - reference epoch are contained in the subframe 1.
-Finally, the only unknown term left in equation
-\eqref{eq:timecorrection2} is $t_{r}$, the relativistic correction
-term. $t_{r}$ can be evaluated by applying the
-equation in \eqref{eq:timecorrection3}. $F$ is a constant
-calculated from the given parameters in \eqref{eq:paramconst1}
-and \eqref{eq:paramconst2}, whereas $e$, $\sqrt{A}$ and $E_{k}$
-are orbit parameters contained in subframe 2 and 3 \citep{GPS-Interface-Specification}.
+coefficients: $a_{f0}$ - \textit{clock offset}, $a_{f1}$ -
+\textit{fractional frequency offset}, $a_{f2}$ - \textit{
+fractional frequency drift}; and
+$t_{0c}$ - \textit{reference epoch} are encapsulated inside
+of subframe 1. Finally, the only unknown term left in equation
+\eqref{eq:timecorrection2} is the \textit{relativistic correction
+term}, denoted as $\Delta t_{r}$. $\Delta t_{r}$ can be evaluated
+by applying the equation given in \eqref{eq:timecorrection3}.
+$F$ is a constant calculated from the given parameters
+in \eqref{eq:paramconst1} and \eqref{eq:paramconst2},
+whereas $e$, $\sqrt{A}$ and $E_{k}$ are \textit{orbit
+parameters} encapsulated in subframe 2 and 3
+\citep{GPS-Interface-Specification}.
\begin{equation}
\label{eq:timecorrection1}
@@ -67,33 +70,75 @@ t=t_{SV}-\Delta t_{SV}
\begin{alignat}{4}
& \Delta t_{SV} &= \;& a_{f0} + a_{f1}(t_{SV}-t_{oc}) + a_{f2}(t_{SV}-t_{oc})^{2} + \Delta t_{r} \label{eq:timecorrection2} \\
& \Delta t_{r} &= \; & Fe\sqrt{A}\sin{E_{k}} \label{eq:timecorrection3} \\
- & F &= \;& \frac{-2\sqrt{\mu}} {c^{2}} = -4.442807633 \cdot 10^{-10} \frac{s}{\sqrt{m}} \label{eq:timecorrection4}
+ & F &= \;& \frac{-2\sqrt{\mu_{e}}} {c^{2}} = -4.442807633 \cdot 10^{-10} \frac{s}{\sqrt{m}} \label{eq:timecorrection4}
\end{alignat}
+However, the broadcast satellite time
+information is not sufficient to estimate the precise
+time at the moment of the signal arival. Even though the signal
+arives in approximately 77 ms, the precision of the atomic clock
+is in the range of 10 ns \citep[Chapter 2]{diggelen2009a-gps}.
+Undoubtedly the signal propagation (travel)
+time, denoted as $t_{prop}$, has to be taken into account.
+Then the exact time at the moment of arival, denoted as
+$t_{exact}$, is given in equation \eqref{eq:exactTime}.
+The signal propagation time must be known to
+estimate the distance from the satellite
+as well as to estimate the position of the GPS receiver.
\begin{equation}
-\label{eq:paramconst1}
- \begin{split}
- \mu = 3.986005\cdot 10^{14} \frac{m^3}{s^2}
- \end{split}
-\quad\Longleftarrow\quad
- \begin{split}
- \mbox{value of Earth's universal gravitational parameters}
- \end{split}
+\label{eq:exactTime}
+t_{exact} = t_{prop}+t
\end{equation}
-
+In order to calculate the signal propagation time between
+the satellite and the receiver, the internal clock
+wave of the of the receiver crystal needs to be
+synchronized with the carrier clock wave
+of the satellite \citep{4560215}. In other words,
+the identical carrier wave replica has to be generated
+on the receiver as on the satellite.
+Due to the nature of wave propagation and various
+errors the signal arives phase disordered at the
+receiver \citep{4560215}.
+The observed phase at the receiver antenna,
+denoted as $\varphi_{o}$, can be described using
+the equation given in \eqref{eq:phaseShift},
+where $\varphi_{GPS}$ represents the known satellite
+carrier wave phase, $\delta \varphi_{SV}$ the clock
+instabilities on the GPS satellite,
+$\varphi_{a}$ the phase shift error
+caused by propagation delays in the ionosphere
+and troposphere respectively and $\delta \varphi_{w}$
+is the wideband noise.
\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}
+\label{eq:phaseShift}
+\varphi_{o} = \varphi_{GPS}+ \delta\varphi_{SV} + \varphi_{a} + \delta \varphi_{w}
\end{equation}
+The task of the syncrhonization process is to
+generate a replica carrier wave with the matching
+phase shift. In the ideal case, the observed phase
+on the antenna and the generated phase on the
+receiver, denoted as $\varphi_{r}$, cancel each other
+out, in other words, equation \eqref{eq:phaseIdealCaset}
+equals to zero.
+\begin{equation}
+\label{eq:phaseIdealCaset}
+\Delta \varphi = \varphi_{o} - \varphi_{r}
+\end{equation}
+\begin{figure}[ht!]
+ \centering
+ \includegraphics[scale=1.0]{img/Phase-Diff.pdf}
+ \caption[]{Two equivalent carrier waves with phase shift}
+\label{img:phaseShift}
+\end{figure}
+If this property is not satisfied, it is not possible
+to demudalte the C/A code from the received signal.
+
+
+More importantly, $t_{exact}$ is used to synchronize various system dependent.
+
\begin{alignat}{4}
& A & = & \; (\sqrt{A})^2 \nonumber \\
@@ -126,6 +171,11 @@ t=t_{SV}-\Delta t_{SV}
\caption[]{Modulation of the GPS signal L1}
\label{img:gpsmod}
\end{figure}
+As seen in \citep{1656803}
+\begin{equation}
+\label{eq:GPSSignalOutput}
+S(t) = \sqrt{\frac{P}{2}}D(t)C(t)cos(2\pi f_{c}+\varphi_{SV}) + n(t)
+\end{equation}
\begin{figure}[ht!]
\centering