From 1960fbb9824f344a31ee876fe48f717ff389e309 Mon Sep 17 00:00:00 2001
From: Markus Bullmann <markus.bullmann@fhws.de>
Date: Wed, 11 Dec 2019 17:38:17 +0100
Subject: [PATCH] Not much DOP

---
 tex/chapters/4_ftmloc.tex | 24 +++++++++++++++++++++---
 1 file changed, 21 insertions(+), 3 deletions(-)

diff --git a/tex/chapters/4_ftmloc.tex b/tex/chapters/4_ftmloc.tex
index 9b5ff62..c159667 100644
--- a/tex/chapters/4_ftmloc.tex
+++ b/tex/chapters/4_ftmloc.tex
@@ -36,10 +36,10 @@ This offset depends on the particular choice of equation to subtract.
 %TODO local minima vs global?
 %TODO was noch?
 
-Another factor which influences the accuracy is the geometry of the setup where the measurements were taken.
+Another factor which influences the localization accuracy is the geometry of the setup where the measurements were taken.
 The accuracy of multilateration estimate depends on the position of the APs and the position of the smartphone relatively to each other.
-Therefore, it is important to consider the actual AP locations for localization which might differ from the AP locations which results the best signal coverage.
-And the walkable area where the localization system should be used.
+Therefore, it is important to consider the actual AP locations for improved localization accuracy and the walkable area where the localization system is used.
+However, the best geometrical setup for localization is not necessarily the best setup for signal coverage.
 
 Best localization results are archived when the distance circles of the APs intersect in a near orthogonal angle.
 Localization performance degrades with wider intersection angles.
@@ -50,6 +50,24 @@ Usually non-optimal AP locations need to be chosen due to environmental constrai
 These geometrical considerations can be founded on geometric dilution of precision (GDOP), which is a indicator which specifies the localization error based on the sender-receiver geometry.
 %DOP ganz nett aber signalstärken spielt auch eine Rolle
 
+\begin{equation}
+A = 
+ \begin{pmatrix}
+  \frac{(x_1 - x)}{R_1} & \frac{(y_1 - y)}{R_1} & \frac{(z_1 - z)}{R_1} \\[0.3em]
+  \frac{(x_2 - x)}{R_2} & \frac{(y_2 - y)}{R_2} & \frac{(z_2 - z)}{R_2} \\[0.3em]
+  \vdots                & \vdots                & \vdots                \\[0.3em]
+  \frac{(x_i - x)}{R_i} & \frac{(y_i - y)}{R_i} & \frac{(z_i - z)}{R_i} \\
+ \end{pmatrix}
+\end{equation}
+where $R_i=\sqrt{(x_i-x)^2+(y_i-y)^2+(z_i-z)^2}$
+
+\begin{equation}
+  Q = (A^TA)^{-1}
+\end{equation}
+
+\begin{equation}
+\text{GDOP} = \sqrt{\text{trace}(Q)}
+\end{equation}
 
 \subsection{Probabilistic}
 %Dichte aus Messungen erzeugen.