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--- a/vorlagen/thesis/src/kapitel_x.tex
+++ b/vorlagen/thesis/src/kapitel_x.tex
@@ -14,11 +14,98 @@ inside the GSM network
\section{GPS Principles}
\begin{figure}[ht!]
\centering
- \includegraphics[scale=0.50]{img/GPS-Modulation.pdf}
+ \includegraphics[scale=0.50]{img/GPS-Principle.pdf}
\caption[]{nanoBTS with its plastic cover. Image courtesy of ip.access ltd}
-\label{img:nanoBTSPlastic}
+\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.
+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
+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
+characterized as bias, drift and aging errors
+\citep{GPS-Interface-Specification}. The correct broadcast
+time can be estimated using the model equation given in
+\ref{eq:timecorrection1} \citep{GPS-Interface-Specification}.
+$t$ is the correctly estimated GPS system time at broadcast
+moment. In equation \ref{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
+can be estimated using predefined equations. The polynomial
+coefficients, a
+Finally, the only unknown term left in equation
+\ref{eq:timecorrection2} is $t_{r}$, the relativistic correction
+term. $t_{r}$ can be evaluated by applying the
+equation in \ref{eq:timecorrection3}.
+
+\begin{equation}
+\label{eq:timecorrection1}
+\centering
+t=t_{SV}-\Delta t_{SV}
+\end{equation}
+
+\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} \label{eq:timecorrection4}
+\end{alignat}
+
+
+\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{figure}[ht!]
+ \centering
+ \includegraphics[scale=0.50]{img/GPS-Modulation.pdf}
+ \caption[]{Modulation of the GPS signal L1}
+\label{img:gpsmod}
+\end{figure}
+
+\begin{figure}[ht!]
+ \centering
+ \includegraphics[scale=0.50]{img/NAV-Message.pdf}
+ \caption[]{One frame of 1500 bits on L1 frequency carrier}
+\label{img:gpsframe}
+\end{figure}
+
+
\chapter{Radio Resource Location Protocol}
\chapter {Working}