some new TeX aspects

tex/bare_conf.tex

@@ -95,6 +95,7 @@
 \newcommand{\percent}{\%}
 \newcommand{\decibel}{dB}
 \newcommand{\dB}{dB}
+\newcommand{\hpa}{hPa}
 \newcommand{\degree}{\ensuremath{^{\circ}}}
 
 % missing math operators

tex/chapters/sensors.tex

@@ -8,7 +8,56 @@
 		route calculation, user checking the route) we use the average of all barometer readings during this
 		timeframe as realtive base $\overline{\mPressure}$.
 		
+		During each transition from $\mStateVec_{t-1}$ to $\mStateVec_t$, the predicted pressure $\mStatePressure$ is
+		adjusted according to the resulting $z$-change, if any:
+		
+		\begin{equation}
+			\mState_{t}^{\mStatePressure} = \mState_{t-1}^{\mStatePressure} + \Delta z \cdot \SI{0.105}{\hpa}
+			,\enskip
+			\Delta z = \mState_{t-1}^{z} - \mState_{t}^z
+			.
+		\end{equation}
+		
 		
 	\subsection{Wi-Fi \& iBeacons}
+		For additional absolute location hints, we use the Smartphones Wi-Fi and iBeacon sensor to measure the signal-strengths
+		of nearby transmitters. As the positions of both \docAP{}s and and \docIBeacon{}s are known beforehand, we compare
+		each measurement with its corresponding signal strength prediction which is defined by the 3D distance $d$
+		and the number of floors $\Delta f$ between the \docAPshort{} and the particle
+		
+		\begin{equation}
+			P_r(d, \Delta f) = \mTXP - 10 \mPLE \log_{10}{\frac{\mMdlDist}{\mMdlDist_0}} + \Delta{f} \mWAF,
+		\end{equation}
+		
+		and calculate the resulting probability as described in \cite{ipin2015}:
+		
+		\begin{equation}
+			\mProb(\mObsVec \mid \mStateVec)_\text{wifi} =
+			\prod\limits_{i=1}^{n} \mathcal{N}(\mRssi_\text{wifi}^{i} \mid P_{r}(\mMdlDist_{i}, \Delta{f_{i}}), \sigma_{\text{wifi}}^2).
+			\label{eq:wifiTotal}
+		\end{equation}
+				
+		For the \docWIFI{} component we thus need two parameters per \docAPshort{}: $\mTXP$ measured at a distance 
+		$\mMdlDist_0$ (usually \SI{1}{\meter}) and the path-loss exponent $\mPLE$ describing the environment.
+		To reduce complexity and system setup time, we use the same values for all \docAP{}s at the cost of accuracy.
+		While, $\mTXP$ is best determined using averaged measurements at a single location,
+		a good estimation of $\mPLE$ requires several measurements and numerical optimization \cite{etwas_aus_der_MA}.
+		$\mPLE$ is thus chosen empirically.
+		
+		For the \docIBeacon{} component we also use \refeq{eq:wifiTotal} but $\mTXP$ is transmitted by each beacon.
+		Again, $\mPLE$ is determined emprically. \todo{faellt hier meist kleiner aus, weil ja kuerzere reichweite etc}
+	
 	
 	\subsection{Step- \& Turn-Detection}
+		
+		To prevent degradation within the particle-filter \cite{??} due to downvoting of particles with increased
+		heading deviation, we incorporate the turn-detection as control-data directly into the transition
+		$p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t-1})$.
+		\cite{thrun?}\cite{lukas2014?} to get a more directed sampling instead of a truly random one.
+		
+		
+		
+		\commentByFrank{todo: wie wird die unsicherheit in der transition eingebracht, sigma, ..} 
+		
+		
+		

tex/chapters/system.tex

@@ -1,7 +1,24 @@
-\section{Indoor Localisation System}
+\section{Recursive Density Estimation}
 
-\subsection{Overview}
-
-\subsection{Transition}
-
-\subsection{Evaluation}
+	\commentByFrank{particle-filter wie bei lukas mit $\vec{o}_{t}$ in transition und $\vec{q}_{t-1}$ in eval??}
+	\commentByFrank{brauchen wir in der observation ueberhaupt noch $q_{t-1}$??}
+	\commentByFrank{das ist die basis fuer unser system}
+	
+	
+	\begin{equation}
+		p(\mStateVec_{t} \mid \langle \mObsVec \rangle_{t}) = \\
+		p(\mObsVec_{t} \mid \mStateVec_{t})
+		\int
+			p(\mStateVec_{t} \mid \mStateVec_{t-1}, \mObsVec_{t})
+			p(\mStateVec_{t-1} \mid \langle \mObsVec \rangle_{t-1}
+			d\mStateVec_{t-1}
+	\end{equation}
+	
+	\begin{equation}
+		\mObsVec = (\mRssiVec_\text{wifi}, \mRssiVec_\text{ib}, \mObsHeading, \mObsPressure)
+	\end{equation}
+	
+	\begin{equation}
+		\mStateVec = (x, y, z, \mObsHeading, \mStatePressure),\enskip
+		x,y,z,\mStatePressure \in \R
+	\end{equation}

tex/misc/functions.tex

@@ -28,7 +28,12 @@
 
 \newcommand{\mMovingAvgWithSize}[1]{\ensuremath{\text{avg}_{#1}}}
 
-\newcommand{\mPressure}{\rho}							% symbol for pressure readings
+\newcommand{\mPressure}{\rho}
+\newcommand{\mObsPressure}{\mPressure_\text{rel}}				% symbol for observation pressure
+\newcommand{\mStatePressure}{\hat{\mPressure}_\text{rel}}		% symbol for state pressure
+\newcommand{\mHeading}{\theta}
+\newcommand{\mObsHeading}{\Delta\mHeading}						% symbol used for the observation heading
+\newcommand{\mStateHeading}{\mHeading}							% symbol used for the state heading
 
 %\newcommand{\docIBeacon}{iBeacon}