2 files changed, 45 insertions, 9 deletions
diff --git a/doc/thesis/chapters/implementation.tex b/doc/thesis/chapters/implementation.tex
index ae4571f..9b8d543 100644
--- a/doc/thesis/chapters/implementation.tex
+++ b/doc/thesis/chapters/implementation.tex
@@ -256,8 +256,9 @@ In this part the fading blocks for the simulation are added. Tow different types
 
 \subsection{Fading with Discrete-time model}
 
-For the statical version according to \ref{sec:discrete-time-model} to implement and illustrat the fading effect, a separate block was created and implemented in the channel. Nearer shown in \ref{lst:fir-block}. This block is based on a FIR filter. It can be displayed with a direct path or without one. With the help of this filter, the delay of the line of side paths are illustrated. In this block it is possible to simulate any number of these paths with different strengths, as long as there is an associated amplitude specified for each delayed path. Unfortunately, these simulation values do not correspond to the realety, because too many incalculable side effects occur, which aren't possiple to ilustrate in this simulation.
-This block was additionally implemented with the method described in \ref{sec:fractional-delay} to allow non-integer delay values compared to the samples shown in \figref{fig:fractional-delay-sinc-plot}. Where the sinc function does not select an integer sample. Which in turn means that the other sampled values do not add up to zero.
-Thus, they will be distributed among the other whole numbers. A window function could also be implemented to limit these values. Here none was implemented because the sinc function is restricted.
+For the statical version according to \ref{sec:discrete-time-model} to implement and illustrat the fading effect, a separate block was created and implemented in the channel. Nearer shown in \ref{lst:fir-block}. This block is based on a FIR filter. It can be displayed with a direct path or without one. With the help of this filter, the delay of the line of side paths are illustrated. In this block it is possible to simulate any number of these paths with different strengths, as long as there is an associated amplitude specified for each delayed path. Unfortunately, these simulation values do not correspond to the realety, because too many incalculable side effects occur, which aren't possiple to ilustrate in this simulation.
+This block was additionally implemented with the method described in \ref{sec:fractional-delay} to allow non-integer delay values compared to the samples shown in \figref{fig:fractional-delay-sinc-plot}. Where the sinc function does not select an integer sample. Which in turn means that the other sampled values do not add up to zero.
+Thus, they will be distributed among the other whole numbers. A window function could also be implemented to limit these values. Here none was implemented because the sinc function is restricted.
 
 \skelpar[5]{
 	Discrabe a perfect plot
@@ -316,17 +317,21 @@ This block was additionally implemented with the method described in \ref{sec:fr
 
 % TODO: Quelle https://ch.mathworks.com/help/comm/ug/fading-channels.html?searchHighlight=rician%20fading&s_tid=srchtitle_rician%2520fading_2#a1070327427b1
 
+In order to represent the effect of the multipaht fading not only statically, a second model was created using the Frequency Selective Fading Model from Gnu Radio, according to \ref{statistical_model}.which was implemented after the algorithm from the paper \cite{Alimohammad2009}. It is based on the sum-of sinusoid principal(SOS)
 
 \begin{german}
-	Um den effect des multipaht fadinngs nicht nur statisch darzu stellen, wurde ein zweites model kreiert mit hilfe des Frequency Selective Fading Models von Gnu Radio, gemäss \ref{statistical_model}.
-	Welcher nach dem Algorthmus aud dem paper \cite{Alimohammad2009} implementiert wurde. 
+	Um den effect des multipaht fadinngs nicht nur statisch darzu stellen, wurde ein zweites model kreiert mit hilfe des Frequency Selective Fading Models von Gnu Radio, gemäss \ref{statistical_model}.Welcher nach dem Algorthmus aud dem paper \cite{Alimohammad2009} implementiert wurde. Er basiert auf dem sum-of sinusoid princip(SOS)
+	
 	Um die resultate einigermassse nach vollziehen zu können wurde ein MATLAP model zur veranschaulichung erstelle.
 	Um ein realistisches beispiel zu haben wurden werte aus dem Skript \cite{Mathis} genomen
 	
 \end{german}
 
 Some realistic value for this block are:
-The first delay when theirs non line of side is zero. The second delayed path depend on the environment of measurement. In an indoor enviroment it is usaely between \(1\cdot10^{-9}\) to \(1\cdot10^{-7}\) and in an outdoor environment between \(1\cdot10^{-7}\) to \(1\cdot10^{-5}\). The rest depends on on the bandwith 
+The first delay when theirs non line of side should be zero. The second delayed path depend on the environment of measurement. In an indoor enviroment it is usually between \(1\cdot10^{-9}\) to \(1\cdot10^{-7}\) and in an outdoor environment between \(1\cdot10^{-7}\) to \(1\cdot10^{-5}\). The rest depends on on the bandwidth. 
+
+
+Rician fading factor K = 0 = Rylehnt Model 
 
 \skelpar[5]{
 	Simulation mit Werten aus dem Skript
diff --git a/doc/thesis/chapters/theory.tex b/doc/thesis/chapters/theory.tex
index 15aa44b..765c4b9 100644
--- a/doc/thesis/chapters/theory.tex
+++ b/doc/thesis/chapters/theory.tex
@@ -231,10 +231,16 @@ An intuitive parameter to quantify how dispersive channel is, is to take the tim
 as is done in \cite{Gallager}. However since in reality some paths get more attenuated than others (\(c_k(t)\) parameters) it also not uncommon to define the delay spread as a weighted mean or even as a statistical second moment (RMS value), where mean tap power \(\expectation\{|c_k(t)|^2\}\) is taken into account \cite{Mathis,Messier}. More sophisticated definitions of delay spread will be briefly mentioned later in section \ref{sec:statistical-model}.
 
 Another important parameter for quantifying dispersion is \emph{coherence bandwidth}, a measure how 
+\skelpar{sentence}
+
+% TODO: End the sentence
 
 \subsection{Effects of multipath fading on modulation constellations}
 
-\skelpar{Beschreiben warnn die Werte hübsch sind}
+% TODO : Can we sai it that way /dose it need to be in the implementation Part?
+
+It is to mention that not every constellation of parameter for a fading illustration leads to a satisfying  plot constellation.
+For example in a Discrete-time Model: the same delay as the samples per Symbol or a multiple of it leads to a special case, where we see the constellation are around the modulate signal points, when there is no line of side path. This is because of \skelpar{Beschreiben warnn die Werte hübsch sind}
 
 \subsection{Discrete-time model} \label{sec:discrete-time-model}
 
@@ -274,10 +280,10 @@ is different from \eqref{eqn:multipath-impulse-response} consider again the plot
 \end{figure}
 
 From a signal processing perspective \eqref{eqn:discrete-multipath-impulse-response} can be interpreted as a simple tapped delay line, schematically drawn in \figref{fig:tapped-delay-line}, which confirms that the presented mathematical model is indeed a FIR filter. Simple multipath channels can be simulated with just a few lines of code, for example the data for the static fading channel in \figref{fig:multipath-frequency-response-plots} is generated in just four lines of Python. The difficulty of fading channels in practice lies in the estimation of the constantly changing parameters \(c_k(t)\) and \(\tau_k(t)\).
-
+%TODO: Code ? 
 
 \subsection{Simulating multipath CIR with FIR filters} \label{sec:fractional-delay}
-% TO Do quelle: http://users.spa.aalto.fi/vpv/publications/vesan_vaitos/ch3_pt1_fir.pdf
+% TODO quelle: http://users.spa.aalto.fi/vpv/publications/vesan_vaitos/ch3_pt1_fir.pdf
 
 \begin{figure}
 	\centering
@@ -327,7 +333,7 @@ Recall that \(h_l(m)\) is a function of time because \(c_k\) and \(\tau_k\) chan
 for some parameter \(\sigma\). Loosely speaking, the distribution needs to be ``circular'' because \(h_l\) is a complex number, which is a ``two dimensional number'', it can however be understood as \(\Re{h_l} \sim \mathcal{N}(0, \sigma^2)\) and \(\Im{h_l} \sim \mathcal{N}(\mu, \sigma^2)\), i.e. having each component being normally distributed.
 
 
-% TO DO : Picture of gaussian distribution
+%TODO : Picture of gaussian distribution
 \begin{subequations}
 	\begin{align}
 		R_{l} (k) &= \E{h_l(m) h_l^*(m+k)}, \\
@@ -352,8 +358,33 @@ for some parameter \(\sigma\). Loosely speaking, the distribution needs to be ``
 	}
 \end{figure}
 
+%TODO :Maby some correction on the descreption, because mentionet earlyer
 \paragraph{NLOS case}
 
+%TODO: Quellen : Skript Mathis und Buch Grundlagen der digitalen Informationsübertagung Peter Adam  Höher
+
+ 
+ In the case of the Rayleight distribution the signal has no line of sight. So to find the probability function of the amplitutes of this superimposition of those infinity of distribute signals:
+\begin{equation} \label{eqn:rayleight fading}
+ 	f(t) = \lim_{N\rightarrow\infty} \frac{1}{\sqrt{N}}\sum_{n=1}^{N} e^{j(\Theta +2\pi jf t)}.
+ \end{equation}  
+whish are nominatet with the factor \(\frac{1}{\sqrt{N}}\) so that the \(\E{|f(t)|²}=1\) and the fact that we are looking at the complex basiband and this prosses are independent and in this the gaussian distribution it can be said that is zero \(\E{f(t)}=0\)
+ 
+
+ It can be explain in two different way with the help of  %TODO: How do you say Quadraturkomponenten in english?
+ quadraturcomponents or the help of the amplitude in time and the associated phase \(\Theta(t) \in[\,0,2\pi)\,\)
+ 
+
+
+
+So it can be said that the amplitude of the rayleightdistribution 
+
+%\begin{equation} \label{eqn:rayleight_fading_probabilety_dencety}
+%	p(a)= 2a \exp{-a^2}
+%
+%\end{equation}
+%TODO: Why not the same as in the skript
+
 \skelpar[4]{Explain statistical model with Rayleighan distribution.}
 \begin{equation}
 	\Re{h_l(n)}, \Im{h_l(n)}