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@@ -415,6 +415,169 @@ Operational &Green - Steady & Default condition if none of the above apply&12 (L
\end {table}
\clearpage
+\section{Carrier wave demodulation}
+\label{sec:carWavDemod}
+The reason why the equivalent carrier wave must be generated is straightforward
+to understand by looking at the multiplication of two sine waves.
+The GPS L1 signal demodulator at the receiver was depicted in figure
+\ref{img:L1Demod}, page \pageref{img:L1Demod}. The incoming signal L1 is multiplied with
+the synthesized sine wave\footnote{Multiplication is the function of
+a mixer, denoted as $\otimes$ in figure \ref{img:L1Demod}.}.
+For the purpose of easier analysis and understanding this concept,
+cosine waves shall be used istead of sine waves. The difference between sine
+and cosine waves is 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 $\cos(A-B)$ together, as respectively
+given in equations \eqref{eq:cos1} and \eqref{eq:cos2}.
+\begin{equation}
+\label{eq:multCosin}
+\cos(A)\cdot\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 $d_{C/A}\cos(\omega_{1}t)$, a similar form is given in equation \eqref{eq:GPSSignalReceived6},
+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),
+$d_{C/A}=d_{PRN}\oplus d_{NAV}$.
+\begin{equation}
+\label{eq:GPSSignalReceived6}
+S(t) = \sqrt{\frac{P}{2}}d_{C/A}cos(2\pi f_{c}+\varphi_{GPS}) + n(t)
+\end{equation}
+If equation \eqref{eq:multCosin} is rewritten with the received GPS signal L1
+and synthesized wave with frequency $f_{2}$ substituted, then the equation results the one
+given in \eqref{eq:cosResult}
+\begin{equation}
+\label{eq:cosResult}
+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 $(\omega_{1}t-\omega_{2}t)$
+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
+specified 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}, 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}
+However, if the frequencies do not match, $f_{1}\neq f_{2}$,
+then the output signal $\frac{1}{2}d_{C/A}$ will be
+modified by the residual frequency $f_{1}-f_{2}$,
+and subsequently this will change the demodulated C/A output (also known as phase shift). Under those circumstances
+the correlator is 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=0.5]{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). Image courtesy of \citep{diggelen2009a-gps}.}
+\label{img:multCAPhase}
+\end{figure}
+
+\clearpage
+\section{C/A wave demodulation}
+\label{sec:CAwaveDemodApend}
+The demodulation process, of finding the correct chipping rate,
+will examined in this appendix section.
+The chipping period $T_{c}$ can be derived from equation \eqref{eq:chipPeriod}.
+The amount of time required to find a matching PRN code shift, $\tau$,
+on the receiverr is proportional to the amount of parallely working LFSRs on the system
+\citep[Chapter 3]{bensky2008wireless}. Clearly with more LFSRs
+the required time for finding the matching phase shift increases.
+\begin{equation}
+\label{eq:chipPeriod}
+T_{c} = \frac{1}{f_{PRN}} = \frac{1}{1.023\cdot 10^6 \mathrm{Hz}}
+\end{equation}
+To determine whether the synthesized PRN code,
+matches the incoming C/A code of the received satellite
+signal, known correlation properties of PRN codes are used,
+as described in section \ref{sec:gpsDataAndSignal}.
+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
+maximum value for the case when each bit from one signal matches
+the bit from the other signal. 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 code produces a
+result of $-3=(+1)\cdot(-1)+(-1)\cdot(+1)+(+1)\cdot(-1)+(+1)\cdot(+1)+(-1)\cdot(+1)$.
+However, 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/Correlation.pdf}
+ \caption{Cross-correlation on three different signals. Image courtesy of \citep{understandGPS}.}
+\label{img:correlatingSignals}
+\end{figure}
+The same principle applies to the transmitted C/A and
+generated PRN code sequences in the GPS receiver. Thus, this can be modeled using
+the equation given in \eqref{eq:autocorrelationProperty},
+where $G_{i}(t)$ is the C/A code\footnote{PRN generated codes for GPS satellites
+are called Gold code sequences since they were first discovered by Dr. Robert Gold.} 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}^{1022} G_{i}(t)G_{i}(t+\tau)d\tau
+\end{equation}
+Another correlation property of the PRN codes is used,
+the fact that in the ideal case the cross-correlation of two
+different PRN codes yields a result of zero. The ideal case of
+PRN code 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}
+$PRN_{i}$ 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 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.
+Once the phase shift, $\tau$, has been found, the C/A code is modulated
+(XORed) with it. The resulting binary code are the transmitted subframes containing data
+required to estimate the position.
+%The implementation problem of finding correct C/A and carrier wave demodulation shall be
+%further explained in the following section \ref{sec:2dSearch}.
+
+\clearpage
\section{GPS assistance data descriptions}
Description of assistance data converted and sent inside the RRLP protocol.
\begin {table}[ht!]