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authorRefik Hadzialic2012-07-04 20:26:27 +0200
committerRefik Hadzialic2012-07-04 20:26:27 +0200
commit513ac23f30ae4acd61b141c64a9eab3bccc7a1c7 (patch)
tree131192815c770dca7476eea9a4e75e6867d14e97
parentWriting (diff)
downloadmalign-513ac23f30ae4acd61b141c64a9eab3bccc7a1c7.tar.gz
malign-513ac23f30ae4acd61b141c64a9eab3bccc7a1c7.tar.xz
malign-513ac23f30ae4acd61b141c64a9eab3bccc7a1c7.zip
Writing and correcting!
-rw-r--r--vorlagen/thesis/maindoc.pdfbin4959107 -> 5010849 bytes
-rw-r--r--vorlagen/thesis/src/bib/literatur.bib7
-rw-r--r--vorlagen/thesis/src/img/BPSKModulation.pdfbin0 -> 43173 bytes
-rw-r--r--vorlagen/thesis/src/img/BPSKModulation.svg234
-rw-r--r--vorlagen/thesis/src/kapitel_x.tex200
5 files changed, 369 insertions, 72 deletions
diff --git a/vorlagen/thesis/maindoc.pdf b/vorlagen/thesis/maindoc.pdf
index b8e1711..b259365 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 8cfc634..ae3f7e8 100644
--- a/vorlagen/thesis/src/bib/literatur.bib
+++ b/vorlagen/thesis/src/bib/literatur.bib
@@ -335,3 +335,10 @@ note = {[Online; accessed 27-June-2012]}
title = {{European Commission Report on Galileo Estimates \$ 1 Trillion in Europe Depends on SatNav}}
}
+@book{9780849316579,
+ Author = {edited by Mohamed Ibnkahla},
+ Title = {Signal Processing for Mobile Communications Handbook},
+ Publisher = {CRC Press},
+ Year = {2004},
+ ISBN = {084931657X}
+} \ No newline at end of file
diff --git a/vorlagen/thesis/src/img/BPSKModulation.pdf b/vorlagen/thesis/src/img/BPSKModulation.pdf
new file mode 100644
index 0000000..8eacee2
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diff --git a/vorlagen/thesis/src/img/BPSKModulation.svg b/vorlagen/thesis/src/img/BPSKModulation.svg
new file mode 100644
index 0000000..b3d549c
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diff --git a/vorlagen/thesis/src/kapitel_x.tex b/vorlagen/thesis/src/kapitel_x.tex
index 0c3b383..5017aa0 100644
--- a/vorlagen/thesis/src/kapitel_x.tex
+++ b/vorlagen/thesis/src/kapitel_x.tex
@@ -13,7 +13,7 @@ inside the GSM network
\chapter{GPS \& Assisted-GPS}
In the new global economy age, GPS positioning has become of important value for various services
and businesses. It has been growing at a rate of 30\% in the past few years and the application
-market is expected to be worth \euro 240 billion by 2020 only in Europe \citep{gpsMoney}.
+market is expected to be worth \euro 240 milliard by 2020 only in Europe \citep{gpsMoney}.
The goal of this chapter is to bring more details and insights of how GPS receivers work.
The chapter is divided in few sections that explain how the data are modulated before transmission,
demodulated on the receiver, how the search space works, how the target user position is estimated and
@@ -39,14 +39,16 @@ this position estimation technique is also known as time-of-arrival (TOA) method
Once sufficient amount of measurements from different satellites were generated,
the position of the receiver can be approximated.
It is important to understand that the positions from the satellites
-need to be known. The general principle of this idea can be seen in figure
-\ref{img:GPSSimplePrinciple}, picture \textit{a)} represents the idea
-with spheres in 3D space and picture \textit{b)} the same idea but in 2D space.
+need to be known and same location reference system has to be used.
+The general principle of this idea can be seen in figure
+\ref{img:GPSSimplePrinciple}, picture (a) represents the idea
+with spheres in 3D space and picture (b) the same idea but in 2D space.
The blue, yellow and green wireframes below the GPS satellites represent the spheres
-for a given range, between the satellite and the GPS user, where the user might be
-located. By intersecting all the three spheres, the position of the user is estimated.
-In the next sections this general idea will be developed in more details
-step by step and the ideas will be verified using the appropriate mathematical
+for a given range, between the satellite and the estimated position of the GPS user
+for the given satellite.
+By intersecting all the three spheres, the position of the user is estimated.
+In the next sections this general idea will be developed in more details,
+step by step, and the ideas will be verified using the appropriate mathematical
models.
\section{GPS data and signal modulation}
@@ -70,15 +72,16 @@ user's position.
\label{img:gpsframe}
\end{figure}
Each subframe can be divided into three fields of data,
-as shown in figure \ref{img:gpssubframe}, telemetry (TLM)
-as well as handover (HOW) word and rest of data.
+as shown in figure \ref{img:gpssubframe}, telemetry (TLM),
+handover (HOW) word and rest of the data (navigation data).
TLM is the first word of the subframe and consists of
a unique preamble used to synchronize and identify
the subframes \citep{9780817643904}. HOW is the second
word of the subframe and consists of the \textit{GPS system time}
and subframe ID \citep{9780817643904}.
GPS system time is the time the atomic clocks on the satellite generate
-at the moment of subframe broadcast \citep{GPS-Interface-Specification}.
+at the moment of subframe broadcast
+and it acts as a time stamp \citep{GPS-Interface-Specification}.
The third segment of the subframe, indicated as rest of data in figure
\ref{img:gpssubframe}, consists of the navigation data. The first subframe
includes data about the satellite accuracy and health as well as parameters
@@ -112,31 +115,77 @@ by the GPS receiver to refine the calculation of delays through
the ionosphere \citep{9780817643904}. The reason why there are 25 frames is because of the last two
subframes, four and five.
Subframes four and five have data which cycle through the 25 frames, i.e. almanac data
-are transmitted for all the GPS satellites in case the receiver found only one satellite
-and once it collected all almanac data, it can precisely search for other GPS satellites.
+are transmitted for all the 24 GPS satellites in case the receiver found only one satellite
+and once it collected all almanac data, it can search for other visible GPS satellites.
These 25 frames create a masterframe. Once the 25 frames
-have been transmitted, the process is repeated from the start.
+have been transmitted, the process is repeated again.
-The just mentioned data are modulated in real time at the moment of transmission.
-The transmitted 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}
-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_{C/A}cos(2\pi f_{c}+\varphi_{SV}) + n(t)
-\end{equation}
+The data are modulated using the binary phase shift keying (BPSK) technique. The
+newly modulated signal is the \textit{L1} signal and it is emitted from the satellite
+directed antennas towards Earth \citep{GPS-Guide}. The BPSK technique works by changing
+the phase of the carrier signal for $180^{\circ}$ at the moment of bit toggle (flipping) in the
+data \citep{GPS-Guide} \citep{9780817643904}.
+Basic principle of this technique can be seen in figure \ref{img:bpskmod}. The carrier wave
+for GPS BPSK modulation is centered at a frequency of 1575.42 MHz \citep{9780817643904}.
+These signals travel an average distance of $20200 \, km$ from the satellite to the GPS receiver
+and are affected by various sources of noise. BPSK modulation is mostly used for satellite links
+because of its simplicity and immunity to noise and signal intereference for the price of
+transfering data at low speed rates \citep[Chapter 1]{9780849316579}. The demodulation process
+of L1 will be discussed and analysed seperately in section \ref{sec:Carrierdemod}.
+\begin{figure}[ht!]
+ \centering
+ \includegraphics[scale=0.50]{img/BPSKModulation.pdf}
+ \caption[]{BPSK Modulation - The top signal is the carrier wave,
+ and it is multiplied with the signal in the middle, which is
+ the data to be transmitted. The resulting signal at the output
+ of the satellite antenna is the third one.}
+\label{img:bpskmod}
+\end{figure}
+
+However, before the raw navigation data enter the BPSK modulation process, they are XORed
+with pseudo random noise (PRN) sequences for different satellites (each satellite
+owns a unique PRN sequence) \citep{9780817643904}.
+The PRN sequences have similar autocorrelation properties as noise, when it is shifted in
+time domain it has a low correlation whereas when it is matched with exact image of itself
+it produces a high correlation peak \citep[Chapter 3]{bensky2008wireless}. This property is used
+for identifying the satellite and for enabling the reception of different data on the same frequency,
+also known as code division multiple access (CDMA). It is important to note that the PRN sequences
+must have a higher frequency than the data, i.e. the bit duration of a PRN sequence is much shorter
+than of the data \citep[Chapter 3]{bensky2008wireless}. Single bits in PRN sequences are called \textit{chips}
+and the complete sequence as \textit{code} \citep[Chapter 3]{bensky2008wireless}. This newly generated
+signal is called direct sequence spread spectrum (DSSS) \citep[Chapter 3]{bensky2008wireless}. In
+GPS terminology it is named as Code/Acquisition (C/A) signal. C/A signal is feed into the BPSK modulation
+process, where it is mixed with the carrier wave and producing the L1 signal. More details will be given in the
+C/A demodulation section \ref{sec:CAdemod}.
+
\begin{figure}[ht!]
\centering
- \includegraphics[scale=0.60]{img/GPS-Modulation.pdf}
+ \includegraphics[scale=0.50]{img/GPS-Modulation.pdf}
\caption[]{Modulation of the GPS signal L1}
\label{img:gpsmod}
\end{figure}
+The described GPS navigation data modulation can be seen in figure \ref{img:gpsmod} and it
+can be represented in form of equation \eqref{eq:GPSSignalReceived1} \citep{1656803}, where $D(t)$
+are the navigation data at the moment $t$, $C(t)$ is the PRN chip at the moment $t$, $cos(2\pi f_{c}+\varphi_{SV})$
+is the generated carrier wave with frequency $f_c$ and phase $\varphi_{SV}$, $P$ is output power of the transmitter
+amplifier.
+\begin{equation}
+\label{eq:GPSSignalReceived1}
+S(t) = PD(t)C(t)cos(2\pi f_{c}+\varphi_{SV})
+\end{equation}
+The equation \ref{eq:GPSSignalReceived1} will be rewritten as given in \ref{eq:GPSSignalReceived2}. It
+represents the same equation but at the GPS receiver after traveling $\approx 20200 \, km$, where $d_{C/A}$
+is the C/A data and $n(t)$ is the random noise at moment $t$ influenced by various factors that influence
+electromagnetic waves.
+\begin{equation}
+\label{eq:GPSSignalReceived2}
+S(t) = \sqrt{\frac{P}{2}}d_{C/A}cos(2\pi f_{c}+\varphi_{SV}) + n(t)
+\end{equation}
+The GPS satellites are positioned in orbits so that at every moment at any spot on Earth, at least four satellites are visible
+(a spot can be considered as a mountain peak since in the cities GPS signals are blocked by buildings).
+In the next section, more details will be revealed on the process of demodulating the GPS L1 signal and acquiring the
+correct time and position.
@@ -177,8 +226,8 @@ 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
+whereas $e$, $\sqrt{A}$ and $E_{k}$ are orbit
+parameters encapsulated in subframe 2 and 3
\citep{GPS-Interface-Specification}.
\begin{equation}
@@ -204,9 +253,9 @@ Undoubtedly the signal propagation (travel)
time, denoted as $t_{prop}$, has to be taken into account.
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
-but is not sufficient to estimate the position of the GPS receiver.
+%The signal propagation time must be known to
+%estimate the distance from the satellite
+%but is not sufficient 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 communication and ranging systems.
@@ -308,7 +357,7 @@ is only in the phase shift, as denoted in equation
\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
+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}
@@ -347,11 +396,19 @@ specified cutoff frequency of the low-pass filter, are cut off by reducing their
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 $\frac{1}{2}d_{C/A}$.
+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 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=0.5]{img/PRN-PhaseShiftAfterDemod.pdf}
@@ -366,18 +423,12 @@ with the DC term (zero frequency producing a constant voltage) leaving only $\fr
will vary as well (third figure).}
\label{img:multCAPhase}
\end{figure}
-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 will change the demodulated C/A output (also known as phase shift). 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}.
+\newpage
\subsection{C/A wave demodulation}
\label{sec:CAdemod}
As a result of the previous step, one can continue with
@@ -409,8 +460,8 @@ This function, $f(\tau)$, generates an PRN code, that is
delayed in phase by $\tau$, where $\tau$ is a multiple of the chipping
rate 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
+The amount of time required to find a matching PRN code shift, $\tau$,
+on the receiverr is proportional to the amount of LFSR on the system
\citep[Chapter 3]{bensky2008wireless}. Clearly with more LFSRs
the required time for finding the matching phase shift increases.
\begin{equation}
@@ -441,7 +492,9 @@ $+5=(+1)\cdot(+1)+(-1)\cdot(-1)+(+1)\cdot(+1)+(+1)\cdot(+1)+(-1)\cdot(-1)$.
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
+where $G_{i}(t)$ is the C/A code\footnote{PRN generated code for GPS satellites
+is called Gold code sequences
+since they were first discovered by Dr. 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}.
@@ -451,8 +504,8 @@ R_{i}(t) = \frac{1}{1023\cdot T_{C/A}} \int_{t=0}^{1023} G_{i}(t)G_{i}(t+\tau)d\
\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},
+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
@@ -476,12 +529,12 @@ further explained in the following section \ref{sec:2dSearch}.
In the following paragraphs an introduction will be given on
the implementation problems of the previously mentioned concepts.
As it can be seen,
-from subsections \ref{sec:CAdemod} and
-\ref{sec:Carrierdemod}, decoding the GPS navigation message is a 2D
+from subsections \ref{sec:Carrierdemod} and
+\ref{sec:CAdemod}, decoding the GPS navigation message is a 2D
search space problem for each GPS satellite
signal acquisition. The 2D search space is limited by well known
physical properties of the GNSS system such as the motion speed of GPS satellites
-and the receiver as well as the frequency oscillator on the receiver.
+(and the receiver) as well as the frequency oscillator on the receiver.
GPS satellites move toward or away
from the GPS receiver with a speed of $800 \, \mathrm{m/s}$
@@ -520,10 +573,10 @@ unknown frequency to be in range of $10 \, \mathrm{kHz}-25 \, \mathrm{kHz}$.
\label{img:prnSearchSpace3d}
\end{figure}
-A typical receiver searches in frequency bands, bins of several hundred Hz regions \citep{1656803}.
+A typical receiver searches in frequency bands (bins) of several hundred Hz \citep{1656803}.
Commonly used frequency bin size is $500 \, \mathrm{Hz}$,
-therefore there are about 20-50 bins to search \citep[Chapter 3]{diggelen2009a-gps}.
-The frequency search bin (band) size is a function of the desired peak magnitude loss (signal to noise ration)
+therefore there are about 20-50 bins to search ($10000\, \mathrm{Hz}/500\, \mathrm{Hz} = 20$) \citep[Chapter 3]{diggelen2009a-gps}.
+The frequency search bin (band) size is a function of the desired peak magnitude loss (signal to noise ratio)
due to the frequency mismatch and integration time period. Larger frequency
bands mean a smaller number of bins to search but
a greater correlation peak magnitude loss.
@@ -535,11 +588,11 @@ where $\Delta f$ is the frequency mismatch in $\mathrm{Hz}$,
in other words it represents the difference
between the received signal frequency and
the synthesized carrier frequency on the receiver;
-and $T_{c}$ is the coherent integration time (usually $0.5\, ms$ according to \citep{implSoftGPSRec}
+and $T_{ci}$ is the coherent integration time (usually $0.5\, ms$ according to \citep{implSoftGPSRec}
and \citep[Chapter 3]{diggelen2009a-gps} but depends on the implementation).
\begin{equation}
\label{eq:mistunigLoss}
-D_{F} = \left\vert \frac{\sin(\pi \Delta fT_{c})}{\pi \Delta fT_{c}} \right\vert
+D_{F} = \left\vert \frac{\sin(\pi \Delta fT_{ci})}{\pi \Delta fT_{ci}} \right\vert
\end{equation}
The frequency mimsmatch loss sinc function, $D_{F}$, is evaluated in dB,
therefore for a loss of $\approx 0.98 \,\mathrm{dB}$, the frequency mismatch ought to be
@@ -583,9 +636,11 @@ The common strategy is to start searching from the middle frequency bins and to
up and down until the entire search space has been exhausted (first 500 Hz,
second -500 Hz, then in the 1000 Hz bin and then in the -1000 Hz bin)
\citep[Chapter 3]{diggelen2009a-gps}.
-This procedure is performed when no extra information are known by the receiver, i.e.
+This procedure is performed when no extra information are known by the receiver (almanac data
+are missing), i.e.
first time the GPS receiver is turned on. It is known under the name of cold start.
-There are three different working mechanisms when it comes to searching
+
+There are three different working modes when it comes to searching
for the GPS satellites. If no information are known,
when some information are known and when almost all information are
known. These three modes are known as cold (as mentioned earlier),
@@ -599,30 +654,30 @@ programmable read only memory (EEPROM). This type of start is known as a warm st
provided that the data in the receivers' EEPROM are not older than 180 days and
its real time clock counter was constantly updated.
In this case, the receiver uses the previously saved information
-to estimate the position of the satellites, therefore the Doppler effects can be estimated.
+to estimate the position of the satellites, therefore the Doppler effects can be roughly estimated.
As a consequence of the known Doppler effect, the frequency bin where to start
-the search first is known as well \citep[Chapter 3]{diggelen2009a-gps}.
-In the same way works the hot start, only the time is precisely
-known in accuracy of submilliseconds.
+the search first is known this time \citep[Chapter 3]{diggelen2009a-gps}.
+In the same way works the hot start, only the ephemeris data and time data are precisely
+known (time ought to be known in accuracy of submilliseconds).
\section{Distance and position estimation}
-This section will focus on examining the distance and position estimation inside of the GPS receiver.
-GPS system, as mentioned earlier, takes advantage of the time of arrival (TOA) ranging concept
+In this section the focus will be on the distance and position estimation inside of the GPS receiver.
+GPS system, as mentioned earlier, takes advantage of the TOA ranging concept
to determine user position. Time is measured how long it takes for a signal to arrive from a
known location.
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.50]{img/Localization.pdf}
- \caption[]{Basic position estimation principle for one satellite}
+ \caption[]{Basic distance estimation principle for one satellite}
\label{img:SatLocalization}
\end{figure}
-In figure \ref{img:SatLocalization} an example concept can be seen where $\vec{u}=(x_u,y_u,z_u)$ represents the
+In figure \ref{img:SatLocalization}, an example concept can be seen, where $\vec{u}=(x_u,y_u,z_u)$ represents the
GPS user position vector with respect to Earth-Centered, Earth-Fixed\footnote{ECEF is a Cartesian coordinate system
where the point $(0,0,0)$ is defined as the center of mass of the Earth \citep{earthCoordinates}.}
(ECEF) coordinate system, $\vec{r}$ is the distance vector from the satellite to the user and $\vec{s}=(x_s,y_s,z_s)$
represents the GPS satellite position with respect to ECEF at a timepoint. Vector $\vec{s}$ is computed from ephemeris data broadcasted
-by the satellite. Distance vector $\vec{r}$, satellite to user, can be computed using equation \eqref{eq:r} and its magnitude is
+by the satellite. The distance vector $\vec{r}$, distance between the satellite and user, can be computed using equation \eqref{eq:r} and its magnitude is
given in equation \eqref{eq:rMag}.
\begin{equation}
\label{eq:r}
@@ -660,7 +715,7 @@ satellite clock from the system time \citep{understandGPS}.
\label{eq:rho}
\rho=r + c(t_{u}-\delta t)
\end{equation}
-Therefore equation \eqref{eq:rMag} can be rewritten as \eqref{eq:rhoR} with respect to equation \eqref{eq:rho}.
+Equation \eqref{eq:rMag} can be rewritten as \eqref{eq:rhoR} with respect to equation \eqref{eq:rho}.
\begin{equation}
\label{eq:rhoR}
\rho - c(t_{u}-\delta t) = \Vert s-u\Vert
@@ -746,6 +801,7 @@ The basic idea of the principle can be seen in figure \ref{img:taylorSeries}.
\label{eq:taylor}
f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + \frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+...
\end{equation}
+
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.50]{img/TaylorSeries.pdf}
@@ -824,9 +880,9 @@ At this step, by solving equation \eqref{eq:MultitaylorFour}, the linearization
\label{eq:SubsTerms2}
\alpha_{xi} = \dfrac{x_i - \hat{x_u}}{\hat{r_i}} \hspace{1.5em} \alpha_{yi} = \dfrac{y_i - \hat{y_u}}{\hat{r_i}} \hspace{1.5em} \alpha_{zi} = \dfrac{z_i - \hat{z_u}}{\hat{r_i}}
\end{equation}
-By rearanging the equation \eqref{eq:MultitaylorDerivAfter}
-and by substituting the terms in \eqref{eq:SubsTerms1} and \eqref{eq:SubsTerms2} into \eqref{eq:MultitaylorDerivAfterRearange},
-the equation \eqref{eq:MultitaylorDerivAfterRearange} resembles the one given in \eqref{eq:userPosition}.
+By rearanging the equation \eqref{eq:MultitaylorDerivAfter} one derives equation \eqref{eq:MultitaylorDerivAfterRearange}.
+And then by substituting the terms in \eqref{eq:SubsTerms1} and \eqref{eq:SubsTerms2} into \eqref{eq:MultitaylorDerivAfterRearange},
+the equation resembles the one given in \eqref{eq:userPosition}.
\begin{equation}
\label{eq:userPosition}
\Delta\rho_i = \alpha_{xi}\Delta x_u + \alpha_{yi}\Delta y_u + \alpha_{zi}\Delta z_u - c\Delta t_u
@@ -912,8 +968,8 @@ off by several seconds and would require additional equipment for synchronizing
\citep{springerlink:10.1007/s10291-002-0028-0}, \citep{901174}. However in CDMA networks the time stamp is
accurate to within $100 \, \mu s$ \citep{springerlink:10.1007/s10291-002-0028-0}. Approximate
location is typically taken to be the location of the BTS from which the target A-GPS receiver
-acquires the assistance data. Ephemeris and navigation data obtained by the A-GPS receiver
-help it to estimate the positions of the satellites and they can greatly
+acquires the assistance data. Ephemeris and navigation data obtained by the A-GPS receiver in the smart phone
+help it to estimate the positions of the GPS satellites. This method can greatly
enhance the sensitivity of the receiver especially in urban environments \citep{springerlink:10.1007/s10291-002-0028-0}.
Conventional GPS receivers require at least up to extra $18$ to $30\,s$ to receive and decode the navigation data