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authorNao Pross <np@0hm.ch>2021-12-15 22:18:40 +0100
committerNao Pross <np@0hm.ch>2021-12-15 22:18:40 +0100
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treefdc04adc574e3e91d544d79a0f4baacbc2858617 /doc
parentRemove geometry frames (diff)
parentlittel changes on the flowgraphs and gui (diff)
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-rw-r--r--doc/thesis/Fading.bib12
-rw-r--r--doc/thesis/chapters/theory.tex23
2 files changed, 30 insertions, 5 deletions
diff --git a/doc/thesis/Fading.bib b/doc/thesis/Fading.bib
index 3d31790..8661615 100644
--- a/doc/thesis/Fading.bib
+++ b/doc/thesis/Fading.bib
@@ -6,7 +6,7 @@
% https://mirror.foobar.to/CTAN/macros/latex/contrib/biblatex/doc/biblatex.pdf
% description for the entry types are in section 2.1 on page 8
-@article{Alimohammad2009,
+@article{M,
title = "Compact Rayleigh and Rician fading simulator based on random walk processes",
author = "Alimohammad, A. and Fard, S.F. and Cockburn, B.F. and Schlegel, C.",
journal = "IET Commun.",
@@ -156,3 +156,13 @@
year = 2011,
language = "de"
}
+
+@BOOK{Hoher2013,
+ title = "Grundlagen der digitalen Informations{\"u}bertragung",
+ author = "Hoher, Peter Adam",
+ publisher = "Springer Vieweg",
+ edition = 2,
+ month = jun,
+ year = 2013,
+ language = "de"
+}
diff --git a/doc/thesis/chapters/theory.tex b/doc/thesis/chapters/theory.tex
index c6f2620..a9bea7d 100644
--- a/doc/thesis/chapters/theory.tex
+++ b/doc/thesis/chapters/theory.tex
@@ -371,16 +371,31 @@ i.e. the amplitude of \(f\) is \emph{Raileigh} distributed.
\label{fig:multipath-statistical-models}
}
\end{figure}
+\skelpar[4]{Explain This formulars}
\paragraph{LOS case}
-In the case of the Rician distribution model. The line of side exist, which means that one of the paths have a straight communication line from the transmitter to the reviser.
-It can be said that a Rayleight distribution is the same as a Rician distribution with a factor K =0.
-For a faktor K= 5.1 the probability function is gaussien distributed.
+In the case of the Ricean distribution model the line of side exist, which means that one of the paths have a straight communication line from the transmitter to the reviser. So there are in addition to the Rayleight model direct components, whish are also gaussian distributed.
+
+\begin{equation} \label{eqn:rician fading}
+ f(t) = \sqrt{\frac{K}{K+1}}+\lim_{N\rightarrow\infty}\frac{1}{\sqrt{K+1}} \frac{1}{\sqrt{N}}\sum_{n=1}^{N} e^{j(\Theta +2\pi jf t)}.
+\end{equation}
+
+The factor \(K\) named Ricean factor it is the ratio of the line of side power to the average power of the distributed components.
+The Phase for the strait line component has no influences for the Random process therefore there set to zero. In the case when \(K = 0 \)
+the Rician distribution becomes a Rayleight distribution on the other hand when \(K\rightarrow \infty \) the distribution becomes an AWGN-channel model (additive white Gaussian noise). When \(K > 0 \) is the phase not equally distributed.
+
+For this distribution model the expectation value for the real part is \(\E{\Re{f(t)}}=\sqrt{\frac{K}{K+1}} \) and for the imaginary part \(\E{\Im{f(t)}}=0\)
+
+So the probability function of the amplitude in this case is:
+\begin{equation} \label{eqn:rician_fading_probabilety_dencety}
+ p(a)= 2a(1+K)\exp{(-K-{a}^2(K+1))}\cdot I_0(2a\sqrt{K(1+K)})
+\end{equation}
+
+Where \(I_0\) the zero ordered modified besselfunction represent.
-\skelpar[4]{Explain statistical model with Rician distribution.}
\begin{equation}
\Re{h_l(n)}, \Im{h_l(n)}
\sim \mathcal{N} \left( \frac{A_l}{\sqrt{2}}, \frac{1}{2} \sigma_l^2 \right)