From 94275811bb308b35732ff3bcfd0aa77e5ebad614 Mon Sep 17 00:00:00 2001
From: Refik Hadzialic
Date: Wed, 13 Jun 2012 00:16:26 +0200
Subject: Writing and pics adding GPS

---
 vorlagen/thesis/src/kapitel_x.tex | 108 ++++++++++++++++++++++++++++++++++----
 1 file changed, 98 insertions(+), 10 deletions(-)

(limited to 'vorlagen/thesis/src/kapitel_x.tex')

diff --git a/vorlagen/thesis/src/kapitel_x.tex b/vorlagen/thesis/src/kapitel_x.tex
index 0a1441a..223e785 100644
--- a/vorlagen/thesis/src/kapitel_x.tex
+++ b/vorlagen/thesis/src/kapitel_x.tex
@@ -19,8 +19,24 @@ inside the GSM network
 \label{img:gpsprinciple}
 \end{figure}
 
+\section{GPS signal modulation}
+\begin{figure}[ht!]
+  \centering
+  \includegraphics[scale=0.50]{img/GPS-Modulation.pdf}
+  \caption[]{Modulation of the GPS signal L1}
+\label{img:gpsmod}
+\end{figure}
 
+\begin{figure}[ht!]
+  \centering
+  \includegraphics[scale=0.50]{img/NAV-Message.pdf}
+  \caption[]{One frame of 1500 bits on L1 frequency carrier}
+\label{img:gpsframe}
+\end{figure}
 
+
+
+\section{GPS signal demodulation}
 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.}
@@ -165,25 +181,97 @@ More importantly, $t_{exact}$ is used to synchronize various system dependent.
  & 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}
+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
+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,
+this is depicted in figure \ref{img:prnCodeCompare} 
+\citep[Chapter 5]{understandGPS}.
 \begin{figure}[ht!]
   \centering
-  \includegraphics[scale=0.50]{img/GPS-Modulation.pdf}
-  \caption[]{Modulation of the GPS signal L1}
-\label{img:gpsmod}
+  \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:prnCodeCompare}
 \end{figure}
-As seen in \citep{1656803}
+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
+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 $T_{c}$
+can be derived from equation \eqref{eq:chipPeriod}.
+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
+the required time for finding the matching phase shift increases.
 \begin{equation}
-\label{eq:GPSSignalOutput}
-S(t) = \sqrt{\frac{P}{2}}D(t)C(t)cos(2\pi f_{c}+\varphi_{SV}) + n(t)
+\label{eq:chipPeriod}
+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
+-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
+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 
+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 
+$+5=(+1)\cdot(+1)+(-1)\cdot(-1)+(+1)\cdot(+1)+(+1)\cdot(+1)+(-1)\cdot(-1)$.
 \begin{figure}[ht!]
   \centering
-  \includegraphics[scale=0.50]{img/NAV-Message.pdf}
-  \caption[]{One frame of 1500 bits on L1 frequency carrier}
-\label{img:gpsframe}
+  \includegraphics[scale=0.50]{img/Correlation.pdf}
+  \caption[]{Cross-correlation on three different signals}
+\label{img:correlatingSignals}
 \end{figure}
-
+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 
+in the auto-correlation function \citep[Chapter 4]{understandGPS}.
+\begin{equation}
+\label{eq:autocorrelationProperty}
+R_{i}(t) = \frac{1}{1023\cdot T_{C/A}} \int_{t=0}^{1023} G_{i}(t)G_{i}(t+\tau)d\tau
+\end{equation}
+Another correlation property of the PRN codes comes in useful,
+the fact that in the ideal case the cross-correlation of two
+different PRN codes yields a result of zero. The ideal case 
+can be modeled as in equation \eqref{eq:prnIdealCaseZero},
+\begin{equation}
+\label{eq:prnIdealCaseZero}
+R_{ij}(\tau) = \int_{-\infty}^{+\infty} PRN_{i}(t)PRN_{j}(t+\tau)d\tau = 0
+\end{equation}
+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
+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.
+
+\section{Distance and position estimation}
 
 \chapter{Radio Resource Location Protocol}
 
-- 
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