summaryrefslogtreecommitdiffstats
diff options
context:
space:
mode:
authorRefik Hadzialic2012-06-10 21:12:35 +0200
committerRefik Hadzialic2012-06-10 21:12:35 +0200
commit0a21193f4114f2c083566a2f8ff3dbfdc52ce034 (patch)
tree6e7168075bd73812b424c90664065a45d4ed5bfc
parentGPS (diff)
downloadmalign-0a21193f4114f2c083566a2f8ff3dbfdc52ce034.tar.gz
malign-0a21193f4114f2c083566a2f8ff3dbfdc52ce034.tar.xz
malign-0a21193f4114f2c083566a2f8ff3dbfdc52ce034.zip
Write
-rw-r--r--vorlagen/thesis/maindoc.pdfbin4303852 -> 4320520 bytes
-rw-r--r--vorlagen/thesis/src/bib/literatur.bib24
-rw-r--r--vorlagen/thesis/src/erklaerung.tex8
-rw-r--r--vorlagen/thesis/src/img/Phase-Diff.pdfbin0 -> 5395 bytes
-rw-r--r--vorlagen/thesis/src/img/Phase-Diff.svg164
-rw-r--r--vorlagen/thesis/src/kapitel_A.tex24
-rw-r--r--vorlagen/thesis/src/kapitel_x.tex138
7 files changed, 310 insertions, 48 deletions
diff --git a/vorlagen/thesis/maindoc.pdf b/vorlagen/thesis/maindoc.pdf
index b981a19..63b2757 100644
--- a/vorlagen/thesis/maindoc.pdf
+++ b/vorlagen/thesis/maindoc.pdf
Binary files differ
diff --git a/vorlagen/thesis/src/bib/literatur.bib b/vorlagen/thesis/src/bib/literatur.bib
index d6f5dc7..6d61c5e 100644
--- a/vorlagen/thesis/src/bib/literatur.bib
+++ b/vorlagen/thesis/src/bib/literatur.bib
@@ -152,3 +152,27 @@ PERFORMANCE STANDARD}",
url = "http://www.gps.gov/technical/ps/2008-SPS-performance-standard.pdf"
}
+@ARTICLE{4560215,
+author={Razavi, A. and Gebre-Egziabher, D. and Akos, D.M.},
+journal={Aerospace and Electronic Systems, IEEE Transactions on}, title={Carrier loop architectures for tracking weak GPS signals},
+year={2008},
+month={april },
+volume={44},
+number={2},
+pages={697 -710},
+keywords={Doppler-aided FLL;Doppler-aided PLL;GPS signals;bit error rate;carrier recovery loop architectures;carrier-to-noise ratio;frequency detector;frequency lock loop;phase lock loop;receiver;Global Positioning System;error statistics;frequency locked loops;phase locked loops;receivers;signal processing;},
+doi={10.1109/TAES.2008.4560215},
+ISSN={0018-9251},}
+
+@INPROCEEDINGS{1656803,
+author={Xuan Guan and DongWei Hu and Jie Chen},
+booktitle={Mobile Technology, Applications and Systems, 2005 2nd International Conference on}, title={Design and implementation of the acquisition circuit in software GPS receiver},
+year={2005},
+month={nov.},
+volume={},
+number={},
+pages={4 pp. -4},
+keywords={Global Positioning System;acquisition circuit;matched filter;power consumption;software GPS receiver;Global Positioning System;matched filters;radio receivers;radiofrequency filters;software radio;},
+doi={10.1109/MTAS.2005.243823},
+ISSN={},}
+
diff --git a/vorlagen/thesis/src/erklaerung.tex b/vorlagen/thesis/src/erklaerung.tex
index e46210c..576b7eb 100644
--- a/vorlagen/thesis/src/erklaerung.tex
+++ b/vorlagen/thesis/src/erklaerung.tex
@@ -21,12 +21,14 @@ Unterschrift \\(Signature)
\section*{\vfill{}
Acknowledgment}
-I would like to thank my supervisors Konrad Meier and Dennis Wehrle for their help
-and support during the thesis work. Beside the help from the supervisors I
+I would like to thank my supervisors Konrad Meier and Dennis Wehrle for their
+encouraging talks during the thesis. Things which have not been done before
+are intellectually seductive in a way. Beside the help from the supervisors I
would like to thank my family and friends who supported me through my master studies, and
the entire Communication systems department for their support, free coffee and
to Prof. Dr. Gerhard Schneider for making all the required hardware available.
I would like to thank Sebastian Schmelzer for
his LaTeX tips, Michael Neves Pereira and Jonathan Bauer for borrowing me their cell
-phones to test my system with and Johan Latocha for patiently explaining me .
+phones to test my system with and Johan Latocha for patiently explaining me words I
+did not understand in the German language.
diff --git a/vorlagen/thesis/src/img/Phase-Diff.pdf b/vorlagen/thesis/src/img/Phase-Diff.pdf
new file mode 100644
index 0000000..669168a
--- /dev/null
+++ b/vorlagen/thesis/src/img/Phase-Diff.pdf
Binary files differ
diff --git a/vorlagen/thesis/src/img/Phase-Diff.svg b/vorlagen/thesis/src/img/Phase-Diff.svg
new file mode 100644
index 0000000..2402843
--- /dev/null
+++ b/vorlagen/thesis/src/img/Phase-Diff.svg
@@ -0,0 +1,164 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!-- Created with Inkscape (http://www.inkscape.org/) -->
+
+<svg
+ xmlns:dc="http://purl.org/dc/elements/1.1/"
+ xmlns:cc="http://creativecommons.org/ns#"
+ xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+ xmlns:svg="http://www.w3.org/2000/svg"
+ xmlns="http://www.w3.org/2000/svg"
+ xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
+ xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
+ width="324.02609"
+ height="214.37007"
+ id="svg2"
+ version="1.1"
+ inkscape:version="0.48.3.1 r9886"
+ sodipodi:docname="Phase-Diff.svg">
+ <sodipodi:namedview
+ id="base"
+ pagecolor="#ffffff"
+ bordercolor="#666666"
+ borderopacity="1.0"
+ inkscape:pageopacity="0.0"
+ inkscape:pageshadow="2"
+ inkscape:zoom="0.98994949"
+ inkscape:cx="-71.94861"
+ inkscape:cy="-43.856121"
+ inkscape:document-units="px"
+ inkscape:current-layer="layer1"
+ showgrid="true"
+ inkscape:window-width="1280"
+ inkscape:window-height="996"
+ inkscape:window-x="0"
+ inkscape:window-y="0"
+ inkscape:window-maximized="1"
+ objecttolerance="1"
+ gridtolerance="1"
+ guidetolerance="1"
+ fit-margin-top="0"
+ fit-margin-left="0"
+ fit-margin-right="0"
+ fit-margin-bottom="0"
+ showguides="true"
+ inkscape:guide-bbox="true">
+ <inkscape:grid
+ type="xygrid"
+ id="grid2985"
+ empspacing="5"
+ visible="true"
+ enabled="true"
+ snapvisiblegridlinesonly="true"
+ units="mm"
+ spacingx="0.1mm"
+ spacingy="0.1mm"
+ originx="-24.802632mm"
+ originy="-159.75mm" />
+ <sodipodi:guide
+ orientation="1,0"
+ position="1.1807438,223.48279"
+ id="guide4884" />
+ </sodipodi:namedview>
+ <defs
+ id="defs4">
+ <marker
+ inkscape:stockid="Arrow2Lend"
+ orient="auto"
+ refY="0"
+ refX="0"
+ id="Arrow2Lend"
+ style="overflow:visible">
+ <path
+ id="path3875"
+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round"
+ d="M 8.7185878,4.0337352 -2.2072895,0.01601326 8.7185884,-4.0017078 c -1.7454984,2.3720609 -1.7354408,5.6174519 -6e-7,8.035443 z"
+ transform="matrix(-1.1,0,0,-1.1,-1.1,0)"
+ inkscape:connector-curvature="0" />
+ </marker>
+ <marker
+ inkscape:stockid="Arrow2Lstart"
+ orient="auto"
+ refY="0"
+ refX="0"
+ id="Arrow2Lstart"
+ style="overflow:visible">
+ <path
+ id="path3872"
+ style="fill-rule:evenodd;stroke-width:0.625;stroke-linejoin:round"
+ d="M 8.7185878,4.0337352 -2.2072895,0.01601326 8.7185884,-4.0017078 c -1.7454984,2.3720609 -1.7354408,5.6174519 -6e-7,8.035443 z"
+ transform="matrix(1.1,0,0,1.1,1.1,0)"
+ inkscape:connector-curvature="0" />
+ </marker>
+ </defs>
+ <metadata
+ id="metadata7">
+ <rdf:RDF>
+ <cc:Work
+ rdf:about="">
+ <dc:format>image/svg+xml</dc:format>
+ <dc:type
+ rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
+ <dc:title></dc:title>
+ </cc:Work>
+ </rdf:RDF>
+ </metadata>
+ <g
+ inkscape:label="Layer 1"
+ inkscape:groupmode="layer"
+ id="layer1"
+ transform="translate(-87.88334,-271.9488)">
+ <path
+ style="fill:none;stroke:#000000;stroke-width:1.77165353;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:none"
+ d="m 88.920675,379.13384 c 107.367645,0 214.735295,0 322.102935,0 m -272.12597,106.29921 0,-212.59842 m -50.128473,73.92751 c 5.967675,1.46367 11.935353,20.05689 17.903023,37.71817 5.96768,17.66128 11.93535,30.5228 17.90303,26.14222 5.96767,-4.38059 11.93535,-25.098 17.90302,-42.03579 5.96768,-16.93779 11.93535,-26.37765 17.90303,-19.19964 5.96767,7.178 11.93535,29.45449 17.90302,45.20677 5.96768,15.75228 11.93535,21.51298 17.90303,11.73335 5.96767,-9.77963 11.93534,-33.00755 17.90302,-47.14463 5.96767,-14.13709 11.93535,-16.0615 17.90302,-3.94701 5.96768,12.11449 11.93535,35.66025 17.90303,47.79652 5.96767,12.13627 11.93535,10.1719 17.90302,-3.94701 5.96768,-14.1189 11.93535,-37.34022 17.90303,-47.14463 5.96767,-9.80441 11.93535,-4.00484 17.90302,11.73335 5.96768,15.73819 11.93535,38.00166 17.90303,45.20677 5.96767,7.20511 11.93535,-2.27147 17.90302,-19.19964 5.96768,-16.92818 11.93535,-37.62651 17.90303,-42.03579 5.96767,-4.40927 11.93535,8.48582 17.90302,26.14222 5.96768,17.65641 11.93535,36.225 17.90303,37.71817 5.96767,1.49317 11.93535,-14.46868 17.90302,-32.3717"
+ title="3*sin(x)"
+ id="path3764"
+ inkscape:connector-curvature="0"
+ sodipodi:nodetypes="cccccsssssssssssssssssc" />
+ <path
+ id="path3818"
+ title="3*sin(x+0.5)"
+ d="m 89.249003,378.07272 c 4.883439,-12.85105 14.081667,-31.24049 20.593517,-31.09755 1.44578,0.0317 2.91753,3.618 4.00177,5.40559 5.96767,9.83892 11.93535,33.08192 17.90302,47.17541 5.96768,14.09349 11.93535,15.92498 17.90303,3.75839 5.96767,-12.16658 11.93535,-35.71207 17.90302,-47.79614 5.96768,-12.08407 11.93535,-10.02682 17.90303,4.13555 5.96767,14.16238 11.93535,37.36809 17.90302,47.11312 5.96768,9.74503 11.93535,3.85516 17.90303,-11.91669 5.96767,-15.77186 11.93535,-38.00481 17.90302,-45.14498 5.96768,-7.14017 11.93535,2.42167 17.90303,19.37278 5.96767,16.95112 11.93535,37.60485 17.90302,41.9454 5.96768,4.34054 11.93535,-8.63244 17.90303,-26.30043 5.96767,-17.668 11.93535,-36.17912 17.90302,-37.60165 5.96768,-1.42253 11.93535,14.60773 17.90303,32.51067 5.96767,17.90293 11.93535,33.76652 17.90302,32.23223 5.96768,-1.5343 11.93535,-20.18457 17.90303,-37.8341 5.96767,-17.64954 11.93535,-30.43287 17.90302,-25.9836 5.96768,4.44927 11.93535,25.21082 17.90303,42.12552"
+ style="fill:none;stroke:#989898;stroke-width:1.77165353;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:1.77165353, 5.3149606;stroke-dashoffset:0"
+ inkscape:connector-curvature="0"
+ sodipodi:nodetypes="csssssssssssssssssc" />
+ <g
+ id="g4872">
+ <path
+ transform="translate(69.980315,271.9488)"
+ inkscape:connector-curvature="0"
+ id="path4848"
+ d="m 85.491072,66.6915 0,4.375"
+ style="fill:none;stroke:#000000;stroke-width:0.35433072;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:none" />
+ <g
+ transform="translate(-0.40886063,-0.08929)"
+ id="g4868">
+ <path
+ style="fill:none;stroke:#000000;stroke-width:0.35433072;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:none;marker-start:url(#Arrow2Lstart);marker-end:url(#Arrow2Lend)"
+ d="m 156.17364,340.60437 19.88829,0"
+ id="path3848"
+ inkscape:connector-curvature="0" />
+ <path
+ style="fill:none;stroke:#000000;stroke-width:0.35433072;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:none"
+ d="m 176.35532,338.72959 0,4.375"
+ id="path4848-9"
+ inkscape:connector-curvature="0" />
+ </g>
+ </g>
+ <text
+ xml:space="preserve"
+ style="font-size:16px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;line-height:125%;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;font-family:Sans;-inkscape-font-specification:Sans"
+ x="157.42073"
+ y="336.37271"
+ id="text4878"
+ sodipodi:linespacing="125%"><tspan
+ sodipodi:role="line"
+ id="tspan4880"
+ x="157.42073"
+ y="336.37271"
+ style="font-size:12px">∆φ</tspan><tspan
+ sodipodi:role="line"
+ x="157.42073"
+ y="351.37271"
+ id="tspan4882" /></text>
+ </g>
+</svg>
diff --git a/vorlagen/thesis/src/kapitel_A.tex b/vorlagen/thesis/src/kapitel_A.tex
index 21f20e4..3e44685 100644
--- a/vorlagen/thesis/src/kapitel_A.tex
+++ b/vorlagen/thesis/src/kapitel_A.tex
@@ -464,4 +464,26 @@ int main(void)
}
\end{lstlisting}
-\section{GPS Constants} \ No newline at end of file
+\section{GPS Constants}
+\begin{equation}
+\label{eq:paramconst1}
+ \begin{split}
+ \mu_{e} = 3.986004418\cdot 10^{14} \frac{m^3}{s^2}
+ \end{split}
+\quad\Longleftarrow\quad
+ \begin{split}
+ \mbox{Geocentric gravitational constant}
+ \end{split}
+\end{equation}
+
+
+\begin{equation}
+\label{eq:paramconst2}
+ \begin{split}
+ c= 2.99792458\cdot 10^{8} \frac{m}{s}
+ \end{split}
+\quad\Longleftarrow\quad
+ \begin{split}
+ \mbox{speed of light}
+ \end{split}
+\end{equation} \ No newline at end of file
diff --git a/vorlagen/thesis/src/kapitel_x.tex b/vorlagen/thesis/src/kapitel_x.tex
index c81182c..0a1441a 100644
--- a/vorlagen/thesis/src/kapitel_x.tex
+++ b/vorlagen/thesis/src/kapitel_x.tex
@@ -19,44 +19,47 @@ inside the GSM network
\label{img:gpsprinciple}
\end{figure}
+
+
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 have precise clocks on board.
+orbiting our planet, at a distance of approximately 20,200 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 clocks generate is called \textit{GPS
-system time}, $t_{SV}$, and it is generated at the moment
-of the frame broadcast \citep{GPS-Interface-Specification}.
-In order to calculate the signal travel time,
-between the satellite and the GPS receiver, their precise
-time difference needs to be known. Therefore, each
-satellite signs the frame before its sent with its exact
-broadcast time. The broadcast time is located in the
+The time the clock generates is called \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}.
+Each satellite signs the frame with its exact
+broadcast time. The broadcast time is encapsulated in the
subframe 1 of the 1500 bit long frame. In addition to the
broadcast time, subframe 1 contains parameters to account
-for the deterministic clock errors emodied in the
-broadcast GPS system time stamp. These errors can be
+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 can be estimated using the model equation given in
+time, denoted as $t$, can be estimated using the model equation given in
\eqref{eq:timecorrection1} \citep{GPS-Interface-Specification}.
-$t$ is the correctly estimated GPS system time at broadcast
-moment. In equation \eqref{eq:timecorrection2}, where the GPS
+In equation \eqref{eq:timecorrection2}, where the GPS
receiver is required to calculate the satellite clock
-offset, $\Delta t_{SV}$, a number of unknown terms can be
-seen. These terms are contained in the subframe 1 or
+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
-coefficients: $a_{f0}$ - clock offset, $a_{f1}$ - fractional
-frequency offset, $a_{f0}$ - fractional frequency drift; and
-$t_{0c}$ - reference epoch are contained in the subframe 1.
-Finally, the only unknown term left in equation
-\eqref{eq:timecorrection2} is $t_{r}$, the relativistic correction
-term. $t_{r}$ can be evaluated by applying the
-equation in \eqref{eq:timecorrection3}. $F$ is a constant
-calculated from the given parameters in \eqref{eq:paramconst1}
-and \eqref{eq:paramconst2}, whereas $e$, $\sqrt{A}$ and $E_{k}$
-are orbit parameters contained in subframe 2 and 3 \citep{GPS-Interface-Specification}.
+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
+\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}.
+$F$ is a constant calculated from the given parameters
+in \eqref{eq:paramconst1} and \eqref{eq:paramconst2},
+whereas $e$, $\sqrt{A}$ and $E_{k}$ are \textit{orbit
+parameters} encapsulated in subframe 2 and 3
+\citep{GPS-Interface-Specification}.
\begin{equation}
\label{eq:timecorrection1}
@@ -67,33 +70,75 @@ t=t_{SV}-\Delta t_{SV}
\begin{alignat}{4}
& \Delta t_{SV} &= \;& a_{f0} + a_{f1}(t_{SV}-t_{oc}) + a_{f2}(t_{SV}-t_{oc})^{2} + \Delta t_{r} \label{eq:timecorrection2} \\
& \Delta t_{r} &= \; & Fe\sqrt{A}\sin{E_{k}} \label{eq:timecorrection3} \\
- & F &= \;& \frac{-2\sqrt{\mu}} {c^{2}} = -4.442807633 \cdot 10^{-10} \frac{s}{\sqrt{m}} \label{eq:timecorrection4}
+ & 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
+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}.
+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}.
+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.
\begin{equation}
-\label{eq:paramconst1}
- \begin{split}
- \mu = 3.986005\cdot 10^{14} \frac{m^3}{s^2}
- \end{split}
-\quad\Longleftarrow\quad
- \begin{split}
- \mbox{value of Earth's universal gravitational parameters}
- \end{split}
+\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 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}.
+The observed phase at the receiver antenna,
+denoted as $\varphi_{o}$, can be described using
+the equation given in \eqref{eq:phaseShift},
+where $\varphi_{GPS}$ represents the known satellite
+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 $\delta \varphi_{w}$
+is the wideband noise.
\begin{equation}
-\label{eq:paramconst2}
- \begin{split}
- c= 2.99792458\cdot 10^{8} \frac{m}{s}
- \end{split}
-\quad\Longleftarrow\quad
- \begin{split}
- \mbox{speed of light}
- \end{split}
+\label{eq:phaseShift}
+\varphi_{o} = \varphi_{GPS}+ \delta\varphi_{SV} + \varphi_{a} + \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
+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.
+\begin{equation}
+\label{eq:phaseIdealCaset}
+\Delta \varphi = \varphi_{o} - \varphi_{r}
+\end{equation}
+\begin{figure}[ht!]
+ \centering
+ \includegraphics[scale=1.0]{img/Phase-Diff.pdf}
+ \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{alignat}{4}
& A & = & \; (\sqrt{A})^2 \nonumber \\
@@ -126,6 +171,11 @@ t=t_{SV}-\Delta t_{SV}
\caption[]{Modulation of the GPS signal L1}
\label{img:gpsmod}
\end{figure}
+As seen in \citep{1656803}
+\begin{equation}
+\label{eq:GPSSignalOutput}
+S(t) = \sqrt{\frac{P}{2}}D(t)C(t)cos(2\pi f_{c}+\varphi_{SV}) + n(t)
+\end{equation}
\begin{figure}[ht!]
\centering