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diff --git a/vorlagen/thesis/src/kapitel_x.tex b/vorlagen/thesis/src/kapitel_x.tex
index 3d3f958..0e02b4a 100644
--- a/vorlagen/thesis/src/kapitel_x.tex
+++ b/vorlagen/thesis/src/kapitel_x.tex
@@ -20,6 +20,19 @@ inside the GSM network
\end{figure}
\section{GPS signal modulation}
+The transmitted signal after the RF frontend is given
+in equation \eqref{eq:GPSSignalReceived} \citep{1656803}.
+\begin{equation}
+\label{eq:GPSSignalReceived}
+S(t) = \sqrt{\frac{P}{2}}D(t)C(t)cos(2\pi f_{c}+\varphi_{SV}) + n(t)
+\end{equation}
+The received signal after the RF frontend is given
+in equation \eqref{eq:GPSSignalReceived} \citep{1656803}.
+\begin{equation}
+\label{eq:GPSSignalReceived}
+S(t) = \sqrt{\frac{P}{2}}d_{C/A}cos(2\pi f_{c}+\varphi_{SV}) + n(t)
+\end{equation}
+
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.50]{img/GPS-Modulation.pdf}
@@ -61,7 +74,7 @@ time, denoted as $t$, can be estimated using the model given in equation
In equation \eqref{eq:timecorrection2}, where the GPS
receiver is required to calculate the satellite clock
offset, denoted as $\Delta t_{SV}$, a number of unknown terms can be
-seen. These terms are encapsulated in the subframe 1. The polynomial
+seen. These terms are encapsulated inside of the transmitted frames. The polynomial
coefficients: $a_{f0}$ - \textit{clock offset}, $a_{f1}$ -
\textit{fractional frequency offset}, $a_{f2}$ - \textit{
fractional frequency drift}; and
@@ -91,8 +104,8 @@ t=t_{SV}-\Delta t_{SV}
Nevertheless, the broadcast satellite time
information is not sufficient to estimate the precise
time at the moment of the signal arival. Even though the signal
-arives in approximately $77 \, ms$\footnote{The propagation time
-depends on user and GPS satellite position.},
+arives in approximately\footnote{Propagation time
+depends on user and GPS satellite position.} $77 \, ms$,
the precision of the atomic clock is in the
range of 10 ns \citep[Chapter 2]{diggelen2009a-gps}.
Undoubtedly the signal propagation (travel)
@@ -101,27 +114,30 @@ In that case, the exact time at the moment of arival is known,
denoted as $t_{exact}$ and is given in equation \eqref{eq:exactTime}.
The signal propagation time must be known to
estimate the distance from the satellite
-as well as to estimate the position of the GPS receiver.
+but is not sufficient to estimate the position of the GPS receiver.
More importantly, $t_{exact}$ time will be later used
to synchronize various time dependent systems like the
-GSM, LTE, GNSS or other localization systems.
+GSM, LTE, GNSS or other communication and ranging systems.
\begin{equation}
\label{eq:exactTime}
t_{exact} = t_{prop}+t
\end{equation}
+
+\subsection{Carrier wave demodulation}
+\label{sec:Carrierdemod}
In order to calculate the signal propagation time between
the satellite and the receiver, the internal sine
-wave synthesizer in the receiver has to be
+wave synthesizer inside of the receiver has to be
synchronized with the carrier sine wave generator
of the GPS satellite \citep{4560215}. In other words,
the identical carrier wave replica has to be generated
on the receiver as on the satellite \citep{736341}.
However, the received signal is not the equivalent
of the transmitted signal. Due to the nature of the
-Doppler effect\footnote{The Doppler effect is a
+Doppler effect\footnote{Doppler effect is a
phenomenon that happens as a result of relative
-motion of the transmitter and
-receiver towards or away from each other and causes the
+motion of the two bodies, transmitter and
+receiver, towards or away from each other and causes
frequency shift of the electromagnetic wave
\citep[Chapter 4]{3540727140}.}
and wave propagation, the transmitted signal arives
@@ -137,6 +153,7 @@ f(t)=\frac{1}{2\pi}\frac{d}{dt}\phi(t)
\label{eq:phaseFreq}
\phi(t) = 2\pi \int_{-\infty}^{t} f(\tau) d\tau
\end{equation}
+
Considering that the GPS satellites orbit the Earth with
a speed of around $3.9 \, km/s$, the Earth rotates
around its axis and the target user
@@ -152,12 +169,12 @@ $\varphi_{a}$ the phase shift error
caused by propagation delays in the ionosphere
and troposphere respectively, $\delta \varphi_{DE}$ the phase shift
caused by the Doppler effect and $\delta \varphi_{w}$
-is the wideband noise.
+is the wideband noise phase shift.
\begin{equation}
\label{eq:phaseShift}
\varphi_{o} = \varphi_{GPS}+ \delta\varphi_{SV} + \varphi_{a} +\delta \varphi_{DE} + \delta \varphi_{w}
\end{equation}
-The task of the syncrhonization process is to
+The task of the demodulation process is to
generate a replica carrier wave with the matching
phase shift and mix it with the incoming signal.
In the ideal case the observed phase
@@ -174,8 +191,8 @@ such that, $\lim \Delta \varphi \approx 0$.
\end{equation}
\begin{figure}[ht!]
\centering
- \includegraphics[scale=1.0]{img/Phase-Diff.pdf}
- \caption[]{Two equivalent carrier waves with phase shift}
+ \includegraphics[scale=0.5]{img/Phase-Diff.pdf}
+ \caption[]{Two equivalent carrier waves with the same frequency but different phase shift}
\label{img:phaseShift}
\end{figure}
\begin{figure}[ht!]
@@ -184,12 +201,13 @@ such that, $\lim \Delta \varphi \approx 0$.
\caption[]{Demodulation of the L1 GPS signal}
\label{img:L1Demod}
\end{figure}
- This is straightforwardly
-understood by looking at the multiplication of two sine waves. The
-GPS L1 signal demodulator at the receiver is depicted in figure
+ This is straightforward to understand by looking at the
+ multiplication of two sine waves. The GPS L1 signal
+ demodulator at the receiver is depicted in figure
\ref{img:L1Demod}, the incoming signal L1 is multiplied with
-the synthesized sine wave (that is the function of a mixer, denoted
-as $\otimes$). For the purpose of easier analysis, cosine waves
+the synthesized sine wave (multiplication is the function of
+a mixer, denoted as $\otimes$ in figure \ref{img:L1Demod}).
+For the purpose of easier analysis, cosine waves
will be used istead of sine waves, the difference between them
is only in the phase shift, as denoted in equation
\eqref{eq:sineEqCosine}.
@@ -202,7 +220,7 @@ can be derived by adding $\cos(A+B)$ and $\cos(A-B)$, as respectively
given in equations \eqref{eq:cos1} and \eqref{eq:cos2}.
\begin{equation}
\label{eq:multCosin}
-\cos(A)\cos(B) = \frac{1}{2}\cos(A-B)+\frac{1}{2}\cos(A+B)
+\cos(A)\cdot\cos(B) = \frac{1}{2}\cos(A-B)+\frac{1}{2}\cos(A+B)
\end{equation}
\begin{equation}
\label{eq:cos1}
@@ -222,7 +240,7 @@ and synthesized wave with a frequency $f_{2}$, the equation results the one
given in \eqref{eq:cosResult}
\begin{equation}
\label{eq:cosResult}
-d_{C/A}\cos(\omega_{1}t)\cos(\omega_{2}t) = \frac{1}{2}d_{C/A}\cos(\omega_{1}t-\omega_{2}t) + \frac{1}{2}d_{C/A}\cos(\omega_{1}t+\omega_{2}t)
+d_{C/A}\cdot\cos(\omega_{1}t)\cos(\omega_{2}t) = \frac{1}{2}d_{C/A}\cdot\cos(\omega_{1}t-\omega_{2}t) + \frac{1}{2}d_{C/A}\cos(\omega_{1}t+\omega_{2}t)
\end{equation}
This leaves the resulting signal with two frequency terms, a low frequency
term $(\omega_{1}t-\omega_{2}t)$
@@ -233,91 +251,75 @@ The high frequency term, $(\omega_{1}+\omega_{2})$, can be filtered out using
a low-pass filter\footnote{A low-pass filter passes
low frequency signals and attenuates
high frequency signals. In other words, signals higher than the
-speciefied cutoff frequency of the low-pass filter, are filtered out by reducing their amplitudes.}.
+speciefied cutoff frequency of the low-pass filter, are cut off by reducing their amplitudes.}.
Ideally, the difference of the angle frequencies is zero,
-as in equation \eqref{eq:delaOmega}, and $\cos(\Delta \omega)=\cos(0)=1$
+as in equation \eqref{eq:delaOmega}, since $\cos(\Delta \omega)=\cos(0)=1$
+and the remaining left signal is only the C/A code multiplied
+with the DC term (zero frequency producing a constant voltage) leaving only $\frac{1}{2}d_{C/A}$.
\begin{equation}
\label{eq:delaOmega}
\Delta \omega = \omega_{1}-\omega_{2} = 0
\end{equation}
-and the remaining left signal is only the C/A code multiplied
-with the DC term (zero frequency) leaving only $\frac{1}{2}d_{C/A}$.
+\begin{figure}[ht!]
+ \centering
+ \includegraphics[scale=0.5]{img/PRN-PhaseShiftAfterDemod.pdf}
+ \caption[]{Effects of the low frequency term on the demodulated output
+ C/A wave on the GPS receiver (the explanations and figures are from top to bottom).
+ If the synthesized frequency is correct, $f_{1}=f_{2}$, the low
+ frequency term becomes a DC term and does not modify the output
+ $d_{C/A}$ wave (first figure). If the frequency matches but the
+ phase not, in this case the phase is shifted for $\pi$, then
+ $d_{C/A}$ is inverted (second figure).
+ If the phase shifts with time, then the amplitude and phase of $d_{C/A}$
+ will vary as well (third figure).}
+\label{img:multCAPhase}
+\end{figure}
However, if the frequencies do not match, $f_{1}\neq f_{2}$,
-then the output signal $\frac{1}{2}d_{C/A}$ may be
+then the output signal $\frac{1}{2}d_{C/A}$ will be
modified by the residual frequency $f_{1}-f_{2}$,
-and subsequently will change (also known as phase shift)
-the demodulated C/A output. Under those circumstances
+and subsequently will change the demodulated C/A output (also known as phase shift). Under those circumstances
the correlator will be unable to match the C/A code with the
correct PRN code. An illustration of this phenomenon is depicted
in figure \ref{img:multCAPhase}.
-\begin{figure}[ht!]
- \centering
- \includegraphics[scale=0.50]{img/PRN-ChipRate.pdf}
- \caption[]{Comparison of original C/A code generated on the
- GPS satellite with two synthesized PRN codes with phase shift on the receiver}
-\label{img:multCAPhase}
-\end{figure}
-\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}
-The received signal after the RF frontend is given
-in equation \eqref{eq:GPSSignalReceived} \citep{1656803}.
-\begin{equation}
-\label{eq:GPSSignalReceived}
-S(t) = \sqrt{\frac{P}{2}}D(t)C(t)cos(2\pi f_{c}+\varphi_{SV}) + n(t)
-\end{equation}
+
+
+
+\subsection{C/A wave demodulation}
+\label{sec:CAdemod}
+As a result of the previous step, one can continue with
+the demodulation of the C/A wave.
Each tracked GPS satellite signal is demodulated seperately
-using the same PRN code, code chipping rate and carrier frequency
-phase for the given satellite \citep[Chapter 4]{understandGPS}.
-The PRN codes for each GPS satellite are well defined and
+using the same PRN code, code chipping rate and carrier frequency-phase
+(which was determined above) for the given satellite
+\citep[Chapter 4]{understandGPS}.
+The PRN codes for each GPS satellite is well defined and
known by the GPS receiver. The receiver has to generate the
same PRN code with matching code chipping rate (phase)
-of the C/A code,
+of the transmitted C/A code,
this is depicted in figure \ref{img:prnCodeCompare}
\citep[Chapter 5]{understandGPS}.
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.50]{img/PRN-ChipRate.pdf}
- \caption[]{Comparison of original C/A code generated on the
- GPS satellite with two synthesized PRN codes with phase shift on the receiver}
+ \caption[]{Comparison between the original C/A code generated on the
+ GPS satellite with two synthesized PRN codes with a different phase shift on the receiver.}
\label{img:prnCodeCompare}
\end{figure}
For the particular example, the matching phase shift was achieved with
the second replica PRN code, with a phase shift of $\tau=0$ but
there could be a case with any other value of $\tau$, $\tau\in[0,1023]$.
-The PRN code synthesizer implementation depends on the GPS receiver
+Implementation of the PRN code synthesizer depends on the GPS receiver
manufacturer but it is usually implemented as a linear feedback shift
registers (LFSR) that produces an output according to a predefined function $f(\tau)$.
-This function generates an PRN code, that is
-delayed in phase by $\tau$, where $\tau$ is a multiple of the chipping period
-$T_{c}=977.5 ns$. The chipping period $T_{c}$
+This function, $f(\tau)$, generates an PRN code, that is
+delayed in phase by $\tau$, where $\tau$ is a multiple of the chipping
+rate period $T_{c}=977.5 \,ns$. The chipping period $T_{c}$
can be derived from equation \eqref{eq:chipPeriod}.
-The time required to find a matching PRN code shift ($\tau$)
+The time required to find a matching PRN code shift, $\tau$,
is proportional to the amount of LFSR on the system
-\citep[Chapter 3]{bensky2008wireless}. Particularly with more LFSRs
+\citep[Chapter 3]{bensky2008wireless}. Clearly with more LFSRs
the required time for finding the matching phase shift increases.
\begin{equation}
\label{eq:chipPeriod}
@@ -333,7 +335,7 @@ a signal is at its maximum if it is in phase, i.e.
summing up the sequence products yields the absolute
maximum value. As an illustration of the idea, an example is
given in figure \ref{img:correlatingSignals}. The cross-correlation
-of the incoming C/A code with the first synthesized PRN produces a
+of the incoming C/A code with the first synthesized PRN code produces a
result of $-3=(+1)\cdot(-1)+(-1)\cdot(+1)+(+1)\cdot(-1)+(+1)\cdot(+1)+(-1)\cdot(+1)$,
whereas the cross-correlation of the incoming C/A code
and the second synthesized PRN code yields a result of
@@ -347,9 +349,9 @@ $+5=(+1)\cdot(+1)+(-1)\cdot(-1)+(+1)\cdot(+1)+(+1)\cdot(+1)+(-1)\cdot(-1)$.
The same principle applies to the sent C/A and
PRN code sequences in the GPS receiver and thus can be modeled using
the equation given in \eqref{eq:autocorrelationProperty},
-where, $G_{i}(t)$ is the C/A code Gold code sequence as a
-function of time, $t$, for the GPS satellite $i$; $T_{C/A}$ is the
-C/A chipping period of $977.5 ns$ and $\tau$ is the phase shift
+where $G_{i}(t)$ is the C/A code Gold code sequence as a
+function of time $t$, for the GPS satellite $i$; $T_{C/A}$ is the
+C/A chipping period of $977.5 \,ns$ and $\tau$ is the phase shift
in the auto-correlation function \citep[Chapter 4]{understandGPS}.
\begin{equation}
\label{eq:autocorrelationProperty}
@@ -366,12 +368,25 @@ R_{ij}(\tau) = \int_{-\infty}^{+\infty} PRN_{i}(t)PRN_{j}(t+\tau)d\tau = 0
where $PRN_{i}$ is the PRN code waveform for GPS satellite $i$ and
$PRN_{j}$ is the PRN code waveform for every other GPS satellite other
than $i$, $i\neq j$ \citep[Chapter 4]{understandGPS}. Equation
-\eqref{eq:prnIdealCaseZero} ``states that the PRN waveforms of satellite
+\eqref{eq:prnIdealCaseZero} ``states that the PRN waveform of satellite
$i$ does not correlate with PRN waveform of any other satellite $j$ for
any phase shift $\tau$'' \citep[Chapter 4]{understandGPS}.
Without the property given in \eqref{eq:prnIdealCaseZero},
the GPS receiver would not be able to smoothly
-differentiate between different GPS satellite signals.
+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 searching problem}
+As it can be seen from the two above subsections, \ref{sec:CAdemod}
+and \ref{sec:Carrierdemod}, decoding the GPS signal is a 2-Dimensional
+searching problem.
+\begin{figure}[ht!]
+ \centering
+ \includegraphics[scale=0.70]{img/2D-SearchSpaceInk.pdf}
+ \caption[]{2D Search space}
+\label{img:prnCodeCompare}
+\end{figure}
+
\section{Distance and position estimation}