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tex/chapters/3_ftm.tex

@@ -147,6 +147,7 @@ After calculating the average ToF the responder transfers the result to the init
 With increasing $n$ the impact of noise is lessened, but the time until the FTM measurement is available for the consuming software increases.
 Therefore, the actual choice of the value of $n$ is a trade-off between precision and measurement delay.
 
+%TODO ToF -> distance ToF/2 * c
 %TODO IEEE 802.11-2016 6.3.58.1
 
 The accuracy of distance estimate depends on the ability of the hardware to detect the line-of-sight signal, or direct path.

tex/chapters/4_ftmloc.tex

@@ -4,6 +4,7 @@
 Bei Indoor Lokalisierung geht es darum eine Position zu ermitteln. Hierfür nutzen wir unterschiedliche Verfahren. namley... usw. 
 
 After measuring several distances to different anchor points one can calculate his current position.
+%TODO Alles mit 2D, weil halt
 
 \subsection{Multilateration}
 %Ganz kurz erläutern was Multilateration eigentlich ist. in 2D min 3 aps und in 3D min. 4D. Aber grundsätzlich gilt: viel hilft viel.
@@ -12,6 +13,20 @@ After measuring several distances to different anchor points one can calculate h
 %FTM Nachteil: Häufig fallen die Messungen aus? Was tun? Alte Werte statisch halten? Keine est berechnen?
 %Conceptually, multilateration determines the position by analytically intersecting at least $3$ circles for a 2-dimensional position, or at least $4$ spheres in case of a 3-dimensional coordinate system.
 
+Each distance measurement $d_i$ constrains the position estimate $\hat{\mPosVec}$ to a circle, where the center of the circle is the known position $\mPosVec_i = (x,y)^T$ of AP $i$.
+Formally the distance is the euclidean distance between the known position and the estimate
+\begin{equation}
+d_i = \| \mPosVec_i - \hat{\mPosVec} \|   \text{.}
+\end{equation}
+In two dimensions three ideal distances form a system of linear equations which can be uniquely solved to obtain the position.
+Given more than three distances no solution can be found which stratifies all the constraints as the linear system is overdetermined.
+Additionally, in the presence of noise and inaccurate measurements an exact analytical solution is not possible.
+In this case an approximative solution $\mPosVec^*$ can be found by using a least squares approach which minimizes the quadratic error between the measured distance and the actual distance at a given point
+\begin{equation}
+  \mPosVec^* = \argmin_{\hat{\mPosVec}} \sum_{i}^{} \left(  \| \mPosVec_i - \hat{\mPosVec} \| - d_i \right)^2  \text{.}
+\end{equation}
+
+
 % each distance defines the position on a circle
 % intersecting the circles gives the position
 % at least 3 circles