Merge branch 'master' of https://github.com/AndreasFMueller/SeminarSpezielleFunktionen

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diff --git a/buch/papers/fm/FM presentation/FM_presentation.pdf b/buch/papers/fm/FM presentation/FM_presentation.pdf
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+++ b/buch/papers/fm/FM presentation/FM_presentation.pdf
diff --git a/buch/papers/fm/FM presentation/FM_presentation.tex b/buch/papers/fm/FM presentation/FM_presentation.tex
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+%% !TeX root = .tex
+
+\documentclass[11pt,aspectratio=169]{beamer}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[ngerman]{babel}
+\usepackage{tikz}
+\usetheme{Hannover}
+
+\begin{document}
+	\author{Joshua Bär}
+	\title{FM - Bessel}
+	\subtitle{}
+	\logo{}
+	\institute{OST Ostschweizer Fachhochschule}
+	\date{16.5.2022}
+	\subject{Mathematisches Seminar - Spezielle Funktionen}
+	%\setbeamercovered{transparent}
+	\setbeamercovered{invisible}
+	\setbeamertemplate{navigation symbols}{}
+	\begin{frame}[plain]
+		\maketitle
+	\end{frame}
+%-------------------------------------------------------------------------------	
+\section{Einführung}
+	\begin{frame}
+		\frametitle{Frequenzmodulation}
+		
+		\visible<1->{
+            \begin{equation} \cos(\omega_c t+\beta\sin(\omega_mt))
+        \end{equation}}
+		
+		\only<2>{\includegraphics[scale= 0.7]{images/fm_in_time.png}}
+		\only<3>{\includegraphics[scale= 0.7]{images/fm_frequenz.png}}
+		\only<4>{\includegraphics[scale= 0.7]{images/bessel_frequenz.png}}
+		
+		
+	\end{frame}
+%-------------------------------------------------------------------------------	
+\section{Proof}
+\begin{frame}
+	\frametitle{Bessel}
+
+	\visible<1->{\begin{align}
+			\cos(\beta\sin\varphi)
+			&=
+			J_0(\beta) + 2\sum_{m=1}^\infty J_{2m}(\beta) \cos(2m\varphi)
+			\\
+			\sin(\beta\sin\varphi)
+			&=
+			J_0(\beta) + 2\sum_{m=1}^\infty J_{2m}(\beta) \cos(2m\varphi)
+			\\
+			J_{-n}(\beta) &= (-1)^n J_n(\beta)
+		\end{align}}
+	\visible<2->{\begin{align}
+			\cos(A + B) 
+			&= 
+			\cos(A)\cos(B)-\sin(A)\sin(B)
+			\\
+			2\cos (A)\cos (B)
+			&=
+			\cos(A-B)+\cos(A+B)
+			\\
+			2\sin(A)\sin(B)
+			&=
+			\cos(A-B)-\cos(A+B)
+		\end{align}}
+\end{frame}
+
+%-------------------------------------------------------------------------------	
+\begin{frame}
+		\frametitle{Prof->Done}
+		\begin{align}
+			\cos(\omega_ct+\beta\sin(\omega_mt))
+			&=
+			\sum_{k= -\infty}^\infty J_{k}(\beta) \cos((\omega_c+k\omega_m)t)
+		\end{align}
+	\end{frame}
+%-------------------------------------------------------------------------------	
+	\begin{frame}
+	\begin{figure}
+		\only<1>{\includegraphics[scale = 0.75]{images/fm_frequenz.png}}
+		\only<2>{\includegraphics[scale = 0.75]{images/bessel_frequenz.png}}
+	\end{figure}
+	\end{frame}
+%-------------------------------------------------------------------------------		
+\section{Input Parameter}
+	\begin{frame}
+	\frametitle{Träger-Frequenz Parameter}
+	\onslide<1->{\begin{equation}\cos(\omega_ct+\beta\sin(\omega_mt))\end{equation}}
+	\only<1>{\includegraphics[scale=0.75]{images/100HZ.png}}
+	\only<2>{\includegraphics[scale=0.75]{images/200HZ.png}}
+	\only<3>{\includegraphics[scale=0.75]{images/300HZ.png}}
+	\only<4>{\includegraphics[scale=0.75]{images/400HZ.png}}
+	\end{frame}
+%-------------------------------------------------------------------------------
+\begin{frame}
+\frametitle{Modulations-Frequenz Parameter}
+\onslide<1->{\begin{equation}\cos(\omega_ct+\beta\sin(\omega_mt))\end{equation}}
+\only<1>{\includegraphics[scale=0.75]{images/fm_3Hz.png}}
+\only<2>{\includegraphics[scale=0.75]{images/fm_5Hz.png}}
+\only<3>{\includegraphics[scale=0.75]{images/fm_7Hz.png}}
+\only<4>{\includegraphics[scale=0.75]{images/fm_10Hz.png}}
+\only<5>{\includegraphics[scale=0.75]{images/fm_20Hz.png}}
+\only<6>{\includegraphics[scale=0.75]{images/fm_30Hz.png}}
+\end{frame}	
+%-------------------------------------------------------------------------------
+\begin{frame}
+\frametitle{Beta Parameter}
+	\onslide<1->{\begin{equation}\sum_{k= -\infty}^\infty J_{k}(\beta) \cos((\omega_c+k\omega_m)t)\end{equation}}
+	\only<1>{\includegraphics[scale=0.7]{images/beta_0.001.png}}
+	\only<2>{\includegraphics[scale=0.7]{images/beta_0.1.png}}
+	\only<3>{\includegraphics[scale=0.7]{images/beta_0.5.png}}
+	\only<4>{\includegraphics[scale=0.7]{images/beta_1.png}}
+	\only<5>{\includegraphics[scale=0.7]{images/beta_2.png}}
+	\only<6>{\includegraphics[scale=0.7]{images/beta_3.png}}
+	\only<7>{\includegraphics[scale=0.7]{images/bessel.png}}
+\end{frame}	
+%-------------------------------------------------------------------------------
+\begin{frame}
+	\includegraphics[scale=0.5]{images/beta_1.png}
+	\includegraphics[scale=0.5]{images/bessel.png}
+\end{frame}	
+\end{document}
diff --git a/buch/papers/fm/FM presentation/README.txt b/buch/papers/fm/FM presentation/README.txt
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+Dies ist die Presentation des FM - Bessel
\ No newline at end of file
diff --git a/buch/papers/fm/FM presentation/images/100HZ.png b/buch/papers/fm/FM presentation/images/100HZ.png
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diff --git a/buch/papers/fm/FM presentation/images/200HZ.png b/buch/papers/fm/FM presentation/images/200HZ.png
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diff --git a/buch/papers/fm/FM presentation/images/300HZ.png b/buch/papers/fm/FM presentation/images/300HZ.png
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+++ b/buch/papers/fm/FM presentation/images/300HZ.png
diff --git a/buch/papers/fm/FM presentation/images/400HZ.png b/buch/papers/fm/FM presentation/images/400HZ.png
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index 0000000..236c428
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+++ b/buch/papers/fm/FM presentation/images/400HZ.png
diff --git a/buch/papers/fm/FM presentation/images/bessel.png b/buch/papers/fm/FM presentation/images/bessel.png
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+++ b/buch/papers/fm/FM presentation/images/bessel.png
diff --git a/buch/papers/fm/FM presentation/images/bessel2.png b/buch/papers/fm/FM presentation/images/bessel2.png
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+++ b/buch/papers/fm/FM presentation/images/bessel2.png
diff --git a/buch/papers/fm/FM presentation/images/bessel_beta1.png b/buch/papers/fm/FM presentation/images/bessel_beta1.png
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+++ b/buch/papers/fm/FM presentation/images/bessel_beta1.png
diff --git a/buch/papers/fm/FM presentation/images/bessel_frequenz.png b/buch/papers/fm/FM presentation/images/bessel_frequenz.png
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diff --git a/buch/papers/fm/FM presentation/images/beta_0.001.png b/buch/papers/fm/FM presentation/images/beta_0.001.png
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index 0000000..7e4e276
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+++ b/buch/papers/fm/FM presentation/images/beta_0.001.png
diff --git a/buch/papers/fm/FM presentation/images/beta_0.1.png b/buch/papers/fm/FM presentation/images/beta_0.1.png
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+++ b/buch/papers/fm/FM presentation/images/beta_0.1.png
diff --git a/buch/papers/fm/FM presentation/images/beta_0.5.png b/buch/papers/fm/FM presentation/images/beta_0.5.png
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+++ b/buch/papers/fm/FM presentation/images/beta_0.5.png
diff --git a/buch/papers/fm/FM presentation/images/beta_1.png b/buch/papers/fm/FM presentation/images/beta_1.png
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+++ b/buch/papers/fm/FM presentation/images/beta_1.png
diff --git a/buch/papers/fm/FM presentation/images/beta_2.png b/buch/papers/fm/FM presentation/images/beta_2.png
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+++ b/buch/papers/fm/FM presentation/images/beta_2.png
diff --git a/buch/papers/fm/FM presentation/images/beta_3.png b/buch/papers/fm/FM presentation/images/beta_3.png
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+++ b/buch/papers/fm/FM presentation/images/beta_3.png
diff --git a/buch/papers/fm/FM presentation/images/fm_10Hz.png b/buch/papers/fm/FM presentation/images/fm_10Hz.png
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index 0000000..51bddc7
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+++ b/buch/papers/fm/FM presentation/images/fm_10Hz.png
diff --git a/buch/papers/fm/FM presentation/images/fm_20hz.png b/buch/papers/fm/FM presentation/images/fm_20hz.png
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diff --git a/buch/papers/fm/FM presentation/images/fm_30Hz.png b/buch/papers/fm/FM presentation/images/fm_30Hz.png
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diff --git a/buch/papers/fm/FM presentation/images/fm_3Hz.png b/buch/papers/fm/FM presentation/images/fm_3Hz.png
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diff --git a/buch/papers/fm/FM presentation/images/fm_40Hz.png b/buch/papers/fm/FM presentation/images/fm_40Hz.png
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diff --git a/buch/papers/fm/FM presentation/images/fm_5Hz.png b/buch/papers/fm/FM presentation/images/fm_5Hz.png
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diff --git a/buch/papers/fm/FM presentation/images/fm_7Hz.png b/buch/papers/fm/FM presentation/images/fm_7Hz.png
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diff --git a/buch/papers/fm/FM presentation/images/fm_frequenz.png b/buch/papers/fm/FM presentation/images/fm_frequenz.png
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diff --git a/buch/papers/fm/FM presentation/images/fm_in_time.png b/buch/papers/fm/FM presentation/images/fm_in_time.png
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