From 14558267799bb552e496a37040ccbd4f00eaa518 Mon Sep 17 00:00:00 2001
From: Refik Hadzialic
Date: Thu, 28 Jun 2012 17:57:10 +0200
Subject: Writing GPS position estimation

---
 vorlagen/thesis/src/kapitel_x.tex | 83 ++++++++++++++++++++++++++++++++++++++-
 1 file changed, 81 insertions(+), 2 deletions(-)

(limited to 'vorlagen/thesis/src/kapitel_x.tex')

diff --git a/vorlagen/thesis/src/kapitel_x.tex b/vorlagen/thesis/src/kapitel_x.tex
index 7d5b479..9acca55 100644
--- a/vorlagen/thesis/src/kapitel_x.tex
+++ b/vorlagen/thesis/src/kapitel_x.tex
@@ -50,6 +50,7 @@ S(t) = \sqrt{\frac{P}{2}}d_{C/A}cos(2\pi f_{c}+\varphi_{SV}) + n(t)
 
 
 \section{GPS signal demodulation}
+\label{sec:SigDemod}
 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.}
@@ -515,13 +516,89 @@ the search first is known as well \citep[Chapter 3]{diggelen2009a-gps}.
 In the same way works the hot start, only the time is precisely
 known in accuracy of submilliseconds. 
 
+
 \section{Distance and position estimation}
 This section will focus on examining the distance and position estimation inside of the GPS system.
 GPS system, as mentioned earlier, takes advantage of the time of arrival (TOA) ranging concept
 to determine user position. Time is measured how long it takes for a signal to arrive from a
 known location. Satellite locations can be estimated using the ephemeris data and the exact time.
-
-
+\begin{figure}[ht!]
+  \centering
+  \includegraphics[scale=0.60]{img/Localization.pdf}
+  \caption[]{}
+\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 
+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 offset 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. Vector $\vec{s}$ is computed from ephemeris data broadcasted
+by the satellite. Vector $\vec{r}$, satellite to user vector, 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}
+Geometric distance $r$ is computed by measuring the signal propagation time, this is illustrated in figure \ref{img:TimingLoc}
+and explained in section \ref{sec:CAdemod}. 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=0.80]{img/TimingLoc.pdf}
+  \caption[]{Estimating the distance by phase shift $\Delta t =t_2 - t_1 =\tau$}
+\label{img:TimingLoc}
+\end{figure}
+\begin{equation}
+\label{eq:rDist}
+r=c\Delta t
+\end{equation}
+Since the clocks are not synchronized, 
+as described in sections \ref{sec:SigDemod} and \ref{sec:2dSearch},
+clock error offsets have to be added to 
+the geometric distance $r$. This new distance is called pseudorange, $\rho$, because the range is 
+determined using the difference of two nonsynchronized clocks (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 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}
+Therefore 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, as mentioned in \ref{sec:SigDemod}
+and for that reason it can be removed (i.e. it is not an unknown term anymore),
+the eqaution \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 the user (GPS receiver) position,
+advance of the receiver clock with respect to the system time , $t_u$, has to be found, i.e. 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 mentioned
+earlier the offset, $t_u$ is also an unknown term, then it is required to have at least four equations \eqref{eq:rhoSats}
+to find all the four unknown terms. As a result of this fact, at least four satellites have to be visible at 
+the same time to estimate the position of the target user. Equation given in \eqref{eq:rhoSats} takes the form given 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}
 \section{Assisted GPS in Wireless networks}
 \label{sec:agps}
 In the following paragraphs Assisted GPS (A-GPS) will be presented and how it works. 
@@ -764,6 +841,8 @@ than 100 m \citep{installnanoBTS}.
 \label{img:connectionDiagram}
 \end{figure}
 
+\chapter{Testing}
+Test if it can be tricked out by the software Dennis mentioned (protect my privacy)!
 \chapter{Implementation}
 
 \chapter{Future work}
-- 
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