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-rw-r--r--vorlagen/thesis/src/kapitel_x.tex190
1 files changed, 146 insertions, 44 deletions
diff --git a/vorlagen/thesis/src/kapitel_x.tex b/vorlagen/thesis/src/kapitel_x.tex
index 223e785..3d3f958 100644
--- a/vorlagen/thesis/src/kapitel_x.tex
+++ b/vorlagen/thesis/src/kapitel_x.tex
@@ -40,11 +40,11 @@ inside the GSM network
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.}
-orbiting our planet, at a distance of approximately 20,200 km,
+orbiting our planet, at a distance of approximately $20200 \, km$,
are equiped with precise atomic clocks \citep[Chapter 2.7]{diggelen2009a-gps}.
These atomic clocks are calibrated and maintained on
-a daily basis by the U.S. Air Force, \citep{GPS-Pentagon}.
-The time the clock generates is called \textit{GPS
+a daily basis by the U.S. Air Force \citep{GPS-Pentagon}.
+The time the atomic clock generates is refered as \textit{GPS
system time}, denoted as $t_{SV}$,
and it is generated as a time stamp at the moment
of the frame broadcast \citep{GPS-Interface-Specification}.
@@ -56,18 +56,17 @@ for the deterministic clock errors embedded in the
broadcasted GPS system time stamp. These errors can be
characterized as bias, drift and aging errors
\citep{GPS-Interface-Specification}. The correct broadcast
-time, denoted as $t$, can be estimated using the model equation given in
+time, denoted as $t$, can be estimated using the model given in equation
\eqref{eq:timecorrection1} \citep{GPS-Interface-Specification}.
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 or they
-can be estimated using predefined equations. The polynomial
+seen. These terms are encapsulated in the subframe 1. The polynomial
coefficients: $a_{f0}$ - \textit{clock offset}, $a_{f1}$ -
\textit{fractional frequency offset}, $a_{f2}$ - \textit{
fractional frequency drift}; and
$t_{0c}$ - \textit{reference epoch} are encapsulated inside
-of subframe 1. Finally, the only unknown term left in equation
+of subframe 1. The only remaining unknown term left in equation
\eqref{eq:timecorrection2} is the \textit{relativistic correction
term}, denoted as $\Delta t_{r}$. $\Delta t_{r}$ can be evaluated
by applying the equation given in \eqref{eq:timecorrection3}.
@@ -89,32 +88,60 @@ t=t_{SV}-\Delta t_{SV}
& F &= \;& \frac{-2\sqrt{\mu_{e}}} {c^{2}} = -4.442807633 \cdot 10^{-10} \frac{s}{\sqrt{m}} \label{eq:timecorrection4}
\end{alignat}
-However, the broadcast satellite time
+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, the precision of the atomic clock
-is in the range of 10 ns \citep[Chapter 2]{diggelen2009a-gps}.
+arives in approximately $77 \, ms$\footnote{The propagation time
+depends on user and GPS satellite position.},
+the precision of the atomic clock is in the
+range of 10 ns \citep[Chapter 2]{diggelen2009a-gps}.
Undoubtedly the signal propagation (travel)
time, denoted as $t_{prop}$, has to be taken into account.
-Then the exact time at the moment of arival, denoted as
-$t_{exact}$, is given in equation \eqref{eq:exactTime}.
+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.
+More importantly, $t_{exact}$ time will be later used
+to synchronize various time dependent systems like the
+GSM, LTE, GNSS or other localization systems.
\begin{equation}
\label{eq:exactTime}
t_{exact} = t_{prop}+t
\end{equation}
In order to calculate the signal propagation time between
-the satellite and the receiver, the internal clock
-wave of the of the receiver crystal needs to be
-synchronized with the carrier clock wave
-of the satellite \citep{4560215}. In other words,
+the satellite and the receiver, the internal sine
+wave synthesizer in 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.
-Due to the nature of wave propagation and various
-errors the signal arives phase disordered at the
-receiver \citep{4560215}.
+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
+phenomenon that happens as a result of relative
+motion of the transmitter and
+receiver towards or away from each other and causes the
+frequency shift of the electromagnetic wave
+\citep[Chapter 4]{3540727140}.}
+and wave propagation, the transmitted signal arives
+phase disordered at the receiver \citep{4560215}.
+This phase disorder is a consequence of the relationship
+between the instantaneous frequency and instantaneous phase
+according to equations \eqref{eq:freqPhase} and \eqref{eq:phaseFreq}.
+\begin{equation}
+\label{eq:freqPhase}
+f(t)=\frac{1}{2\pi}\frac{d}{dt}\phi(t)
+\end{equation}
+\begin{equation}
+\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
+with the GPS receiver may move as well, the Doppler effect
+is unavoidable.
The observed phase at the receiver antenna,
denoted as $\varphi_{o}$, can be described using
the equation given in \eqref{eq:phaseShift},
@@ -123,22 +150,27 @@ carrier wave phase, $\delta \varphi_{SV}$ the clock
instabilities on the GPS satellite,
$\varphi_{a}$ the phase shift error
caused by propagation delays in the ionosphere
-and troposphere respectively and $\delta \varphi_{w}$
+and troposphere respectively, $\delta \varphi_{DE}$ the phase shift
+caused by the Doppler effect and $\delta \varphi_{w}$
is the wideband noise.
\begin{equation}
\label{eq:phaseShift}
-\varphi_{o} = \varphi_{GPS}+ \delta\varphi_{SV} + \varphi_{a} + \delta \varphi_{w}
+\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
generate a replica carrier wave with the matching
-phase shift. In the ideal case, the observed phase
+phase shift and mix it with the incoming signal.
+In the ideal case the observed phase
on the antenna and the generated phase on the
-receiver, denoted as $\varphi_{r}$, cancel each other
-out, in other words, equation \eqref{eq:phaseIdealCaset}
-equals to zero.
+receiver, denoted as $\varphi_{rec}$, cancel each other
+out, that is to say, equation \eqref{eq:phaseIdealCase}
+equals zero. The circuit responsible for generating the same
+carrier wave is the phase locked loop (PLL).
+The PLL modifies the synthesized wave parameters
+such that, $\lim \Delta \varphi \approx 0$.
\begin{equation}
-\label{eq:phaseIdealCaset}
-\Delta \varphi = \varphi_{o} - \varphi_{r}
+\label{eq:phaseIdealCase}
+\Delta \varphi = \varphi_{o} - \varphi_{rec}
\end{equation}
\begin{figure}[ht!]
\centering
@@ -146,15 +178,85 @@ equals to zero.
\caption[]{Two equivalent carrier waves with phase shift}
\label{img:phaseShift}
\end{figure}
-If this property is not satisfied, it is not possible
-to demudalte the C/A code from the received signal.
-
-
-
-
-
-More importantly, $t_{exact}$ is used to synchronize various system dependent.
-
+\begin{figure}[ht!]
+ \centering
+ \includegraphics[scale=0.5]{img/L1-Demodulation.pdf}
+ \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
+\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
+will be used istead of sine waves, the difference between them
+is only in the phase shift, as denoted in equation
+\eqref{eq:sineEqCosine}.
+\begin{equation}
+\label{eq:sineEqCosine}
+\sin(\pm x) = \cos\bigg(\frac{\pi}{2} \pm x\bigg)
+\end{equation}
+Multiplication of two cosine waves, as in equation \eqref{eq:multCosin},
+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)
+\end{equation}
+\begin{equation}
+\label{eq:cos1}
+\cos(A+B) = \cos(A)\cos(B)-\sin(A)\sin(B)
+\end{equation}
+\begin{equation}
+\label{eq:cos2}
+\cos(A-B) = \cos(A)\cos(B)+\sin(A)\sin(B)
+\end{equation}
+The incoming GPS L1 signal with a frequency $f_{1}$, given in figure \ref{img:L1Demod},
+can be written as $d_{C/A}\cos(\omega_{1}t)$, where $\omega_{1}=2\pi f_{1}$ is
+the angle frequency and
+$d_{C/A}$ is the C/A data (navigation message modulated with the PRN code),
+$d_{C/A}=d_{PRN}\oplus d_{NAV}$.
+If equation \eqref{eq:multCosin} is rewritten with the received GPS signal L1
+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)
+\end{equation}
+This leaves the resulting signal with two frequency terms, a low frequency
+term $(\omega_{1}t-\omega_{2}t)$
+and a high frequency term $(\omega_{1}t+\omega_{2}t)$,
+the $t$ can be taken in front of the bracket as it
+is a common multiplier.
+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.}.
+Ideally, the difference of the angle frequencies is zero,
+as in equation \eqref{eq:delaOmega}, and $\cos(\Delta \omega)=\cos(0)=1$
+\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 $\frac{1}{2}d_{C/A}$.
+However, if the frequencies do not match, $f_{1}\neq f_{2}$,
+then the output signal $\frac{1}{2}d_{C/A}$ may 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
+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 \\
@@ -223,9 +325,9 @@ T_{c} = \frac{1}{f_{PRN}} = \frac{1}{1.023\cdot 10^6}
\end{equation}
To determine whether the synthesized PRN code,
-matches the incoming C/A code from the satellite,
-known correlation properties of PRN codes are used.
-Since the signal is modeled as a sequence of +1's and
+matches the incoming C/A code of the received satellite
+signal, known correlation properties of PRN codes are used.
+Since the PRN code is modeled as a sequence of +1's and
-1's, the autocorrelation of
a signal is at its maximum if it is in phase, i.e.
summing up the sequence products yields the absolute
@@ -265,11 +367,11 @@ 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
-$i$ does not correlate with PRN waveform of any other satellite for
+$i$ does not correlate with PRN waveform of any other satellite $j$ for
any phase shift $\tau$'' \citep[Chapter 4]{understandGPS}.
-Without this
-property, the GPS receiver would not be able to smoothly
-differentiate between best phase shifts.
+Without the property given in \eqref{eq:prnIdealCaseZero},
+the GPS receiver would not be able to smoothly
+differentiate between different GPS satellite signals.
\section{Distance and position estimation}