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-rw-r--r--vorlagen/thesis/src/kapitel_A.tex75
1 files changed, 39 insertions, 36 deletions
diff --git a/vorlagen/thesis/src/kapitel_A.tex b/vorlagen/thesis/src/kapitel_A.tex
index 1148f79..6ce6cc0 100644
--- a/vorlagen/thesis/src/kapitel_A.tex
+++ b/vorlagen/thesis/src/kapitel_A.tex
@@ -14,10 +14,11 @@
\end{itemize}
\addchap{Appendix}
-\numberwithin{equation}{subsection}
-\numberwithin{table}{subsection}
-\captionsetup[figure]{list=no}
-\captionsetup[table]{list=no}
+\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}
@@ -711,10 +712,10 @@ Idot&Rate of inclination angle (semicircles/second)
\clearpage
\section{GPS distance and position estimation}
\label{sec:distanceAndPosition}
-In this section the focus is set on distance and position estimation inside of the GPS receiver.
+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 position. Time is measured how long it takes for a signal to arrive from a
-known location.
+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}
@@ -725,9 +726,11 @@ In figure \ref{img:SatLocalization}, an example concept can be seen, where $\vec
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 timepoint. Vector $\vec{s}$ is computed from ephemeris data broadcasted
-by the satellite. The distance vector $\vec{r}$, distance between the satellite and user, can be computed using equation \eqref{eq:r} and its magnitude is
-given in equation \eqref{eq:rMag}.
+represents the GPS satellite position with respect to ECEF at a timepoint $t_1$.
+$t_1$ is the timepoint 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}
@@ -736,9 +739,9 @@ given in equation \eqref{eq:rMag}.
\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}
-and it was discussed 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
+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!]
@@ -751,13 +754,11 @@ geometric distance $r$ is computed, as given in equation \eqref{eq:rDist}.
\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 \textit{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) that generate PRN codes\footnote{pseudo - Not genuine; sham; not perfect.}. 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{System
+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}
@@ -769,14 +770,14 @@ Equation \eqref{eq:rMag} can be rewritten as \eqref{eq:rhoR} with respect to equ
\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 discussed in \ref{sec:SigDemod}
-and for that reason it can be removed for sake of simplicity, i.e. it is not an unknown term anymore,
-then the eqaution \eqref{eq:rhoR} can be rewritten as \eqref{eq:rhoNew}.
+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 the user (GPS receiver) position,
+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}
@@ -786,10 +787,10 @@ has to be solved, where $i$ is the index of visible satellites at the moment of
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 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 in
+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.
+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
@@ -799,20 +800,22 @@ 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 than
+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 linearization method\footnote{Linear approximation is a technique where a function
is approximated using a linear function.}
-shall be presented to find the unknown terms $(x_u,y_u,z_u,t_u)$, i.e. out of an approximate position and clock
-offset the true user position and the true clock offset shall be calculated.
+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}
-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}. Suppose that the approximation of the
+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}.
@@ -821,7 +824,7 @@ can be rewritten as an approximate pseudorange \eqref{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,
-shall be expressed by the approximate values and an incremental component as shown in equation \eqref{eq:userCoordinates} \citep{understandGPS}.
+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}
@@ -837,10 +840,10 @@ as in \eqref{eq:rhoSatsNewFunwithApprox}.
\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\footnote{Taylor
+In the next step the pseudorange function shall be approximated using Taylor series (linearization 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}.} (linearization of the nonlinear equation). Taylor
+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)