summaryrefslogtreecommitdiffstats
path: root/vorlagen/thesis/src/kapitel_x.tex
diff options
context:
space:
mode:
Diffstat (limited to 'vorlagen/thesis/src/kapitel_x.tex')
-rw-r--r--vorlagen/thesis/src/kapitel_x.tex25
1 files changed, 13 insertions, 12 deletions
diff --git a/vorlagen/thesis/src/kapitel_x.tex b/vorlagen/thesis/src/kapitel_x.tex
index 4da0984..39b9cad 100644
--- a/vorlagen/thesis/src/kapitel_x.tex
+++ b/vorlagen/thesis/src/kapitel_x.tex
@@ -524,7 +524,7 @@ to determine user position. Time is measured how long it takes for a signal to a
known location.
\begin{figure}[ht!]
\centering
- \includegraphics[scale=0.60]{img/Localization.pdf}
+ \includegraphics[scale=0.50]{img/Localization.pdf}
\caption[]{Basic position estimation principle for one satellite}
\label{img:SatLocalization}
\end{figure}
@@ -550,7 +550,7 @@ propagation time. By multiplying the propagation time, $\Delta t$, with the spee
geometric distance $r$ is computed, as given in equation \eqref{eq:rDist}.
\begin{figure}[ht!]
\centering
- \includegraphics[scale=0.80]{img/TimingLoc.pdf}
+ \includegraphics[scale=0.50]{img/TimingLoc.pdf}
\caption[]{Estimating the distance by phase shift $\Delta t =t_2 - t_1 =\tau$}
\label{img:TimingLoc}
\end{figure}
@@ -606,7 +606,8 @@ equation\footnote{Nonlinear
equations, also known as polynomial equations, are equations that cannot 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 possible easily to find an explicit solution to nonlinear as for linear equations.
+It is not straightforward to find explicit solutions of nonlinear equations, it is more difficult than
+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.}
@@ -689,7 +690,7 @@ and $\mathbf{a}=(\hat{x_u},\hat{y_u},\hat{z_u},\hat{t_u})$.
\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}) \\[0.3em]
+ 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
@@ -782,14 +783,14 @@ with the inverse term of $\boldsymbol{\alpha}$, it yields the result of the unkn
\label{eq:userPositionMatrixFinal}
\Delta \boldsymbol{x} = \boldsymbol{\alpha}^{-1} \Delta\boldsymbol{\rho}
\end{equation}
-The
-
-
-
-
-
-
-
+Linearization is repeated in a loop, where in the next round the approximate positions are set
+to the just derived position value, 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 itterations \citep{pseudorangeError}.
+Risks exist that the solutions will 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}.
\section{Assisted GPS in Wireless networks}