summaryrefslogtreecommitdiffstats
path: root/vorlagen/thesis/src/kapitel_x.tex
diff options
context:
space:
mode:
authorRefik Hadzialic2012-06-19 16:45:39 +0200
committerRefik Hadzialic2012-06-19 16:45:39 +0200
commitc9d089f392d54cfb9534d1bd1d61a0e23a60a98a (patch)
tree696254d9a6cb60e4e8603e7d4ec27861fb860e2b /vorlagen/thesis/src/kapitel_x.tex
parentWriting stuff (diff)
downloadmalign-c9d089f392d54cfb9534d1bd1d61a0e23a60a98a.tar.gz
malign-c9d089f392d54cfb9534d1bd1d61a0e23a60a98a.tar.xz
malign-c9d089f392d54cfb9534d1bd1d61a0e23a60a98a.zip
Start to write about A-GPS
Diffstat (limited to 'vorlagen/thesis/src/kapitel_x.tex')
-rw-r--r--vorlagen/thesis/src/kapitel_x.tex97
1 files changed, 85 insertions, 12 deletions
diff --git a/vorlagen/thesis/src/kapitel_x.tex b/vorlagen/thesis/src/kapitel_x.tex
index 27e222f..ad939fd 100644
--- a/vorlagen/thesis/src/kapitel_x.tex
+++ b/vorlagen/thesis/src/kapitel_x.tex
@@ -376,47 +376,120 @@ the GPS receiver would not be able to smoothly
differentiate between different GPS satellite signals.
Once the phase shift, $\tau$, has been found, the C/A code is modulated
(XORed) with it. The resulting binary code will be the navigation message.
-\section{2-Dimensional search space problem}
+The implementation problem of finding correct C/A and carrier wave demodulation will be
+further explained in the following section \ref{sec:2dSearch}.
+
+\subsection{Implementation of the 2D search space problem}
+\label{sec:2dSearch}
As it can be seen, from subsections \ref{sec:CAdemod} and
\ref{sec:Carrierdemod}, decoding the GPS navigation message is a 2D
search space problem for each GPS satellite
signal acquisition. The 2D search space is limited by well known
-physical properties of the GNSS system such as the motion speed of GPS satellites,
-GPS receiver and receiver oscillator. GPS satellites move toward or away
+physical properties of the GNSS system such as the motion speed of GPS satellites
+and the receiver as well as the frequency oscillator on the receiver.
+
+GPS satellites move toward or away
from the GPS receiver with a speed of $800 \, \mathrm{m/s}$
\citep[Chapter 3]{diggelen2009a-gps}. The Doppler effect on the frequency
of the satellite can be estimated using equation \eqref{eq:dopplerEffectSpeed},
-where $f_{e}$ is the emitting frequency (L1), $v_{SV}$ the speed of the
-satellite and $c$ is the speed of light.
+where $f_{e}$ is the emitting frequency (L1), $v_{SV}$ is the speed of the
+satellite towards (away from) the receiver and $c$ is the speed of light.
\begin{equation}
\label{eq:dopplerEffectSpeed}
f_{DE} = f_{e}\frac{v_{SV}}{c}
\end{equation}
Inserting the appropriate values in equation \eqref{eq:dopplerEffectSpeed}
yields a result of $\approx4.2 \, \mathrm{kHz}$, for $800 \, \mathrm{m/s}$ and
-$-4.2 \, \mathrm{kHz}$ (if the satellite moves away from the GPS receiver
+$\approx-4.2 \, \mathrm{kHz}$ (if the satellite moves away from the GPS receiver
then the speed is taken as negative). This makes a range of $\approx8.4 \mathrm{kHz}$.
The Doppler effect of the GPS receiver motion can be ignored since for
each $1 \, \mathrm{km/h}$ of movement, it affects the frequency
range for $\approx 1.46 \mathrm{Hz}$.
+
On the other hand, the frequency offset induced by the reference
oscillator in the GPS receiver can not be ignored.
The frequency search space is ``additionaly affected for $1.575 \, \mathrm{kHz}$
of unknown frequency offset for each $1 \, \mathrm{ppm}$
(\textit{parts per million}) of the unknown receiver
-oscillator offset'' \citep[Chapter 4]{understandGPS}. The reference oscillators
+oscillator offset'' \citep[Chapter 3]{diggelen2009a-gps}. The reference oscillators
in GPS receivers have typically an offset of
$\pm0.5, \pm1, \pm2, \pm3, \mathrm{or} \pm5 \,\mathrm{ppm}$
-\citep{daishinku}, \citep[Chapter 4]{understandGPS}, the standard in
-smart phone design has been set to $\pm 2.5 \mathrm{ppm}$ \citep{oscillatorGPSSmarthPhone}.
+\citep{daishinku}, \citep[Chapter 3]{diggelen2009a-gps}, the standard in
+smart phone design has been set to $\pm 2.5 \, \mathrm{ppm}$
+\citep{oscillatorGPSSmarthPhone}. In the worst case this makes the
+unknown frequency to be in range of $10 \, \mathrm{kHz}-25 \, \mathrm{kHz}$.
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.70]{img/2D-SearchSpaceInk.pdf}
- \caption[]{Part of frequency/code delay search space for a single GPS satellite}
-\label{img:prnCodeCompare}
+ \caption[]{Segment of the frequency/code delay search space for a single GPS satellite}
+\label{img:prnSearchSpace3d}
\end{figure}
-
+A typical receiver searches in frequency bands, bins of several hundred Hz regions \citep{1656803}.
+Commonly used frequency bin size is $500 \, \mathrm{Hz}$,
+therefore there are about 20-50 bins to search \citep[Chapter 3]{diggelen2009a-gps}.
+The frequency search bin (band) size is a function of the desired peak magnitude loss (signal to noise ration)
+due to the frequency mismatch and integration time period. Larger frequency
+bands mean a smaller number of bins to search but
+a greater correlation peak magnitude loss.
+The frequency search bin size can be
+estimated using the frequency
+mimsmatch loss sinc function given in equation \eqref{eq:mistunigLoss} \citep{implSoftGPSRec},
+\citep[Chapter 6]{diggelen2009a-gps},
+where $\Delta f$ is the frequency mismatch in $\mathrm{Hz}$,
+in other words it represents the difference
+between the received signal frequency and
+the synthesized carrier frequency on the receiver;
+and $T_{c}$ is the coherent integration time (usually $0.5\, ms$ according to \citep{implSoftGPSRec}
+and \citep[Chapter 3]{diggelen2009a-gps} but depends on the implementation).
+\begin{equation}
+\label{eq:mistunigLoss}
+D_{F} = \left\vert \frac{\sin(\pi \Delta fT_{c})}{\pi \Delta fT_{c}} \right\vert
+\end{equation}
+The frequency mimsmatch loss sinc function, $D_{F}$, is evaluated in dB,
+therefore for a loss of $\approx 0.98 \,\mathrm{dB}$, the frequency mismatch ought to be
+$\Delta f = 250\, \mathrm{Hz}$,
+due to the fact that the maximum loss will occur when the frequency is differing
+by 1/2 of the bin spacing. That is to say, for a bin space of 500 Hz, it is 250 Hz.
+
+``The total range of possible GPS code delays is $1\, ms$. This is because the GPS C/A
+PRN code is $1 \,ms$ long, and then it repeats. The PRN code chipping rate is $1.023
+\,\mathrm{MHz}$, and there are 1023 chips in the complete $1\, ms$ epoch'' \citep[Chapter 3]{diggelen2009a-gps}.
+
+%Size of the frequency
+%bin is inversely proportional to the ratio between the amplitude of the detected
+%peak and other non-peak values,
+%the smaller the bins are the higher the peak will be.
+
+For the purpose of better understanding, a segment of the
+frequency/code delay search space is shown in figure \ref{img:prnSearchSpace3d}.
+The peak implies the correct frequency and code delay have been found. In figure
+\ref{img:prnSearchSpace3d} smaller frequency bins have been used so that the concept
+becomes understandable to the reader.
+
+The speed of searching the 2D search space (finding the peak)
+depends on the complexity and strategy of the
+implemented algorithm \citep[Chapter 6]{9780817643904}. In the worst case,
+there are in total 102300 conbinations in the search space,
+this can be derived from equation \eqref{eq:totalSearch}.
+\begin{equation}
+\label{eq:totalSearch}
+\mathrm{Search \, Space} = 50 \,\mathrm{(bins)} \cdot 1023\, \mathrm{(C/A \,codes)} \cdot 2\, \mathrm{(Phases\, per\, C/A\, chip)}
+\end{equation}
+The common strategy is to start searching from the middle frequency bin,
+first 500 Hz, second -500 Hz, then 1000 Hz and -1000 Hz until the entire
+search space has been exhausted \citep[Chapter 3]{diggelen2009a-gps}.
+This search space can be reduced by changing the sensitivity of the GPS receiver with the already given
+equation \eqref{eq:mistunigLoss} or delivering required information to the GPS receiver like the frequency
+ranges, phase-shifts and etc. This method is also known as A-GPS \citep{755159} and will be further analysed
+in the following subsection.
+
+\subsection{The A in A-GPS}
+After the peaks have been found for each seen satellite, it can
+receive the navigation messages and estimate the position.
+There are three different searching modes, if no information are known,
+when some information are known and when almost all information are
+known. These three modes are known as cold, warm and hot start.
\section{Distance and position estimation}
\chapter{Radio Resource Location Protocol}