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} -- cgit v1.2.3-55-g7522